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Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension

Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface... 1IntroductionThis article is concerned with the stability of equilibrium figure of uniformly rotating viscous incompressible fluid in R3{{\mathbb{R}}}^{3}with surface tension, where the equilibrium figure is rotationally symmetric about a certain axis. The fluid occupies a region Ω(t)\Omega \left(t)at time t≥0t\ge 0, which is surrounded by a free interface Γ(t)\Gamma \left(t). We denote the initial position of Γ(t)\Gamma \left(t)by Γ0{\Gamma }_{0}. Besides, we denote the velocity and pressure of the fluid by v(x,t)v\left(x,t)and π(x,t)\pi \left(x,t), respectively, and the unit outward normal on Γ(t)\Gamma \left(t)by νΓ{\nu }_{\Gamma }. The normal velocity and the doubled mean curvature of Γ(t)\Gamma \left(t)with respect to νΓ{\nu }_{\Gamma }are denoted by VΓ{V}_{\Gamma }and HΓ{{\mathscr{H}}}_{\Gamma }, respectively. Then, the motion of the fluid is governed by the following system: (1.1)∂tv+(v⋅∇)v−μΔv+∇π=0,in Ω(t),divv=0,in Ω(t),S(v,π)νΓ=σHΓνΓ,on Γ(t),VΓ=v⋅νΓ,on Γ(t),v(0)=v0,in Ω(0),Γ(0)=Γ0.\left\{\begin{array}{ll}{\partial }_{t}v+\left(v\cdot \nabla )v-\mu \Delta v+\nabla \pi =0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(t)\text{},\\ {\rm{div}}\hspace{0.33em}v=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(t)\text{},\\ S\left(v,\pi ){\nu }_{\Gamma }=\sigma {{\mathscr{H}}}_{\Gamma }{\nu }_{\Gamma },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\Gamma \left(t)\text{},\\ {V}_{\Gamma }=v\cdot {\nu }_{\Gamma },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\Gamma \left(t)\text{},\\ v\left(0)={v}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(0)\text{},\\ \Gamma \left(0)={\Gamma }_{0}.& \end{array}\right.In this article, Ω(0)\Omega \left(0)is assumed to be bounded. Here, S(v,π)S\left(v,\pi )is the stress tensor defined by S(v,π)≔μ(∇v+[∇v]⊤)−πIin Ω(t),S\left(v,\pi ):= \mu \left(\nabla v+{\left[\nabla v]}^{\top })-\pi I\hspace{1.0em}\hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(t)\text{},where μ\mu is a positive constant and M⊤{{\mathsf{M}}}^{\top }means the transpose of M{\mathsf{M}}. Without loss of generality, we may assume that external (atmospheric) pressure is zero in this article.It is well known that the Navier-Stokes equations admits the following stationary solution that corresponds to a rigid rotation: v∞(x)=ωe3×x,π∞(x)=ω22∣x′∣2+p0,{v}_{\infty }\left(x)=\omega {e}_{3}\times x,\hspace{1.0em}{\pi }_{\infty }\left(x)=\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}+{p}_{0},where ω∈R\omega \in {\mathbb{R}}describes a constant angular velocity and p0{p}_{0}is some constant, and we have set x′=(x1,x2,0)∈R3x^{\prime} =\left({x}_{1},{x}_{2},0)\in {{\mathbb{R}}}^{3}. Substituting (v∞,π∞)\left({v}_{\infty },{\pi }_{\infty })into the boundary condition (1.1)3{}_{3}, we obtain the equation for the doubled mean curvature HΓ∞{{\mathscr{H}}}_{{\Gamma }_{\infty }}of Γ∞{\Gamma }_{\infty }: (1.2)σHΓ∞+ω22∣x′∣2+p0=0onΓ∞≔∂Ω∞.\sigma {{\mathscr{H}}}_{{\Gamma }_{\infty }}+\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}+{p}_{0}=0\hspace{1.0em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\hspace{0.33em}{\Gamma }_{\infty }:= \partial {\Omega }_{\infty }.This defines the equilibrium figure Ω∞{\Omega }_{\infty }of the rotating liquid. Notice that, if Ω∞{\Omega }_{\infty }is rotationally symmetric about the x3{x}_{3}axis, the boundary condition (1.1)4{}_{4}is automatically satisfied. In the sequel, Ω∞{\Omega }_{\infty }is assumed to be rotationally symmetric with respect to the axis defined by e3{e}_{3}. Obviously, in the case ω=0\omega =0, we observe the classical Young-Laplace law σHΓ∞+p0=0\sigma {{\mathscr{H}}}_{{\Gamma }_{\infty }}+{p}_{0}=0. However, our interest here is to investigate the case ω≠0\omega \ne 0and, especially, characterize the stability of the equilibrium figure by means of the second variation of the energy functional instead of a restriction on the value of ω\omega .We note that a solution to (1.1) satisfies the following equalities: (1.3)∣Ω(t)∣=∣Ω(0)∣,∫Ω(t)v(x,t)dx=∫Ω(0)v0(x)dx,∫Ω(t)(v×x)dx=∫Ω(0)(v0×x)dx.| \Omega \left(t)| =| \Omega \left(0)| ,\hspace{1.0em}\mathop{\int }\limits_{\Omega \left(t)}v\left(x,t){\rm{d}}x=\mathop{\int }\limits_{\Omega \left(0)}{v}_{0}\left(x){\rm{d}}x,\hspace{1.0em}\mathop{\int }\limits_{\Omega \left(t)}\left(v\times x){\rm{d}}x=\mathop{\int }\limits_{\Omega \left(0)}\left({v}_{0}\times x){\rm{d}}x.By passing a uniformly moving coordinate system x^=x−V^0t\widehat{x}=x-{\widehat{V}}_{0}tand v^=v−V^0\widehat{v}=v-{\widehat{V}}_{0}, where V^0=∣Ω(0)∣−1∫Ω(0)v0dx{\widehat{V}}_{0}=| \Omega \left(0){| }^{-1}{\int }_{\Omega \left(0)}{v}_{0}{\rm{d}}x, and by rotating coordinate axes, we suppose that (1.4)∫Ω(t)v(x,t)dx=∫Ω(0)v0(x)dx=0,∫Ω(t)(v×x)dx=∫Ω(0)(v0×x)dx=γe3≔γ(0,0,1),∫Ω(0)xdx=0.\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{\Omega \left(t)}v\left(x,t){\rm{d}}x& =& \mathop{\displaystyle \int }\limits_{\Omega \left(0)}{v}_{0}\left(x){\rm{d}}x=0,\\ \mathop{\displaystyle \int }\limits_{\Omega \left(t)}\left(v\times x){\rm{d}}x& =& \mathop{\displaystyle \int }\limits_{\Omega \left(0)}\left({v}_{0}\times x){\rm{d}}x=\gamma {e}_{3}:= \gamma \left(0,0,1),\\ \mathop{\displaystyle \int }\limits_{\Omega \left(0)}x{\rm{d}}x& =& 0.\end{array}It follows form the Reynolds transport theorem that (1.4)3{\left(1.4)}_{3}gives (1.5)∫Ω(t)xdx=∫0tdds∫Ω(s)xdxds=∫0t∫Ω(s)vdxds=0,\mathop{\int }\limits_{\Omega \left(t)}x{\rm{d}}x=\underset{0}{\overset{t}{\int }}\frac{{\rm{d}}}{{\rm{d}}s}\left(\mathop{\int }\limits_{\Omega \left(s)}x{\rm{d}}x\right){\rm{d}}s=\underset{0}{\overset{t}{\int }}\left(\mathop{\int }\limits_{\Omega \left(s)}v{\rm{d}}x\right){\rm{d}}s=0,which means that the barycenter point of the domain Ω(t)\Omega \left(t)is always suited at the origin. Finally, to guarantee that (1.4)2{\left(1.4)}_{2}holds with v=v∞v={v}_{\infty }and Ω(t)=Ω∞\Omega \left(t)={\Omega }_{\infty }, the angular velocity ω\omega and the value γ\gamma should satisfy the relation (1.6)−ω∫Ω∞∣x′∣2dx=γ.-\omega \mathop{\int }\limits_{{\Omega }_{\infty }}| x^{\prime} {| }^{2}{\rm{d}}x=\gamma .Notice that the value ω\omega should be determined from a given quantity γ\gamma . Namely, Γ∞{\Gamma }_{\infty }is determined from γ\gamma , where Γ∞{\Gamma }_{\infty }is a smooth solution to (1.2) subject to (1.6). If ∣γ∣≪1| \gamma | \ll 1, there exists a unique Γ∞{\Gamma }_{\infty }satisfying (1.2) and (1.6), see Solonnikov [30, Thm. 5.1] and Watanabe [40, Prop. A.1]. Finally, the multiplication of (1.2) by x⋅νΓ∞x\cdot {\nu }_{{\Gamma }_{\infty }}and integration over Γ∞{\Gamma }_{\infty }leads to the expression for p0{p}_{0}, where νΓ∞{\nu }_{{\Gamma }_{\infty }}is the unit outward normal on Γ∞{\Gamma }_{\infty }. In fact, it follows from (1.2) that ∫Γ∞σHΓ∞x⋅νΓ∞dΓ∞+∫Γ∞ω22∣x′∣2x⋅νΓ∞dΓ∞+∫Γ∞p0x⋅νΓ∞dΓ∞=0.\mathop{\int }\limits_{{\Gamma }_{\infty }}\sigma {{\mathscr{H}}}_{{\Gamma }_{\infty }}x\cdot {\nu }_{{\Gamma }_{\infty }}{\rm{d}}{\Gamma }_{\infty }+\mathop{\int }\limits_{{\Gamma }_{\infty }}\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}x\cdot {\nu }_{{\Gamma }_{\infty }}{\rm{d}}{\Gamma }_{\infty }+\mathop{\int }\limits_{{\Gamma }_{\infty }}{p}_{0}x\cdot {\nu }_{{\Gamma }_{\infty }}{\rm{d}}{\Gamma }_{\infty }=0.The relation HΓ∞νΓ∞=ΔΓ∞x{{\mathscr{H}}}_{{\Gamma }_{\infty }}{\nu }_{{\Gamma }_{\infty }}={\Delta }_{{\Gamma }_{\infty }}x, x∈Γ∞x\in {\Gamma }_{\infty }, and the divergence theorem imply −σ∫Γ∞∣∇Γ∞x∣2dΓ∞+5ω22∫Ω∞∣x′∣2dx+3∣Ω∞∣p0=0.-\sigma \mathop{\int }\limits_{{\Gamma }_{\infty }}| {\nabla }_{{\Gamma }_{\infty }}x{| }^{2}{\rm{d}}{\Gamma }_{\infty }+\frac{5{\omega }^{2}}{2}\mathop{\int }\limits_{{\Omega }_{\infty }}| x^{\prime} {| }^{2}{\rm{d}}x+3| {\Omega }_{\infty }| {p}_{0}=0.Hence, we deduce that p0=2σ∣Γ∞∣3∣Ω∞∣+5γω6∣Ω∞∣.{p}_{0}=\frac{2\sigma | {\Gamma }_{\infty }| }{3| {\Omega }_{\infty }| }+\frac{5\gamma \omega }{6| {\Omega }_{\infty }| }.Notice that, by (1.6), we see that p0{p}_{0}is strictly positive.The free boundary problem of (1.1) is said to be finding a family of hypersurfaces {Γ(t)}t≥0{\left\{\Gamma \left(t)\right\}}_{t\ge 0}and appropriately smooth solutions vvand π\pi . Notice that finding a family of hypersurfaces {Γ(t)}t≥0{\left\{\Gamma \left(t)\right\}}_{t\ge 0}is equivalent to finding a family of {Ω(t)}t≥0{\left\{\Omega \left(t)\right\}}_{t\ge 0}. Since many authors have considered problems similar to (1.1) in various settings, we only mention the details of those papers that dealt with the effect of surface tension included on the free interface (the case σ>0\sigma \gt 0) and with assuming that Ω(0)\Omega \left(0)is bounded. For the case of surface tension on the free boundary, we refer the reader to papers [1,3,4, 5,11,17, 18,35,36, 37] that deal with the case where the initial domain Ω(0)\Omega \left(0)is an infinite layer of finite depth with a rigid bottom, see also a recent article by Saito and Shibata [23] that dealt with the case of the bottomless ocean. When Ω(0)\Omega \left(0)is bounded, the first contribution to the solvability of (1.1) traces back to a long series of papers by Solonnikov [29,30,31]. Specifically, Solonnikov investigated the problem in L2{L}^{2}regularity framework, i.e., he showed the local existence and uniqueness of solutions for (1.1) in Sobolev-Slobodetskiĭ spaces W22+α,1+α2{W}_{2}^{2+\alpha ,1+\frac{\alpha }{2}}with 1/2<α<11\hspace{0.1em}\text{/}\hspace{0.1em}2\lt \alpha \lt 1. To obtain the local-in-time solutions in Hölder and anisotropic Sobolev regularity frameworks, we refer to the works by Moglilevskiĭ and Solonnikov [16] and Shibata [24], respectively.To be precise, Shibata [24] obtained the local existence result for uniformly C3{C}^{3}-domains Ω(0)\Omega \left(0)such that the weak Dirichlet problem is uniquely solvable on W^01,q(Ω(0))≔{φ∈Llocq(Ω(0))∣∇φ∈Lq(Ω(0))3,φ∣Γ0=0}{\widehat{W}}_{0}^{1,q}\left(\Omega \left(0)):= \left\{\varphi \in {L}_{{\rm{loc}}}^{q}\left(\Omega \left(0))| \nabla \varphi \in {L}^{q}{\left(\Omega \left(0))}^{3},\varphi {| }_{{\Gamma }_{0}}=0\right\}, 1<q<∞1\lt q\lt \infty (cf. [27, Def. 3.2.5]).The unique global existence theorem in the L2{L}^{2}regularity frameworks was proved by Solonnikov [30], where the solution converges to a uniform rigid rotation of the liquid about a certain axis, provided that the initial velocity and the initial angular momentum (i.e., ∣γ∣| \gamma | ) are sufficiently small and Γ0{\Gamma }_{0}is sufficiently close to a sphere. The similar result was also established in Hölder regularity framework, see Padula and Solonnikov [19]. More recently, the author [40] extends the result obtained in [19] the class of anisotropic Sobolev spaces Wp,q2,1{W}_{p,q}^{2,1}with 2<p<∞2\lt p\lt \infty and 3<q<∞3\lt q\lt \infty satisfying 2/p+3/q<12\hspace{0.1em}\text{/}p+3\text{/}\hspace{0.1em}q\lt 1, which can also be regarded as an extension of Shibata [26] dealing with the case ω=0\omega =0. Notice that, in the previous studies [16,30,40], the equilibrium figure is uniquely determined by the constant γ\gamma . However, it was necessary to assume that ∣γ∣| \gamma | is small since this assumption yields the smallness of ∣ω∣| \omega | , so that we could find a unique smooth solution to equation (1.2) subject to (1.6) based on a standard contraction mapping theorem; see Solonnikov [30, Thm. 5.1] and Watanabe [40, Prop. A.1]. On the other hand, Solonnikov [34] showed that the smallness condition on ∣γ∣| \gamma | can be replaced by the condition of the positivity of the second variation of the functional Eω{E}_{\omega }given by (1.7)Eω(h)=∫Γ(t)σdΓ−∫Ω(t)ω22∣x′∣2dx−∫Ω(t)p0dx,{E}_{\omega }\left(h)=\mathop{\int }\limits_{\Gamma \left(t)}\sigma {\rm{d}}\Gamma -\mathop{\int }\limits_{\Omega \left(t)}\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}{\rm{d}}x-\mathop{\int }\limits_{\Omega \left(t)}{p}_{0}{\rm{d}}x,provided that there exists a smooth surface Γ∞{\Gamma }_{\infty }such that Γ∞{\Gamma }_{\infty }satisfies (1.2) and (1.6) and is rotationally symmetric about the x3{x}_{3}axis. In particular, he showed that the stability result can be obtained by the positivity of the second variation of the energy functional Eω(h){E}_{\omega }\left(h)within the Hölder regularity framework.The aim of this article is to extend the aforementioned Hölder regularity result obtained by Solonnikov [34, Thm. 2.1] in the Lp{L}^{p}-in-time and Lq{L}^{q}-in-space (Lp−Lq{L}^{p}-{L}^{q}) setting with 2<p<∞2\lt p\lt \infty and 3<q<∞3\lt q\lt \infty satisfying 2/p+3/q<12\hspace{0.1em}\text{/}p+3\text{/}\hspace{0.1em}q\lt 1, which provides an optimal regularity on the initial data. In particular, in contrast to Shibata [26], we investigate the stability of nontrivial stationary solutions (i.e., (v∞,π∞,Γ∞)\left({v}_{\infty },{\pi }_{\infty },{\Gamma }_{\infty })with ω≠0\omega \ne 0) to the three-dimensional Navier-Stokes equations with surface tension. To prove our main result, we transform the system (1.1) into a problem on a domain F{\mathscr{F}}surrounded by a fixed surface G{\mathscr{G}}. To observe smoothing of the unknown interface, we rely on the direct mapping method via the Hanzawa transform, i.e., we approximate the free surface Γ(t)\Gamma \left(t)by a real analytic hypersurface G{\mathscr{G}}, in terms of the Hausdorff distance of the second-order normal bundles being as small as we wish. As the transformed problem becomes a quasilinear parabolic type PDE with inhomogeneous boundary data, it is widely known that the key idea to show the local-in-time existence of regular solution is to use the maximal Lp−Lq{L}^{p}-{L}^{q}regularity result for an associated linearized problem. In this article, however, we will address the global existence issue, and hence, it is also required to derive some decay property of an associated linearized system – combining the local existence result and the decay property yields the global existence result due to a standard argument. To this end, we use the maximal Lp−Lq{L}^{p}-{L}^{q}regularity result for the shifted linearized system to rewrite the linearized system as an abstract evolution equation with homogeneous boundary data. Since the shifted linearized system admits a unique solution in the maximal Lp−Lq{L}^{p}-{L}^{q}regularity class that decays exponentially, it suffices to study the decay property for a solution to the abstract evolution equation. Although Solonnikov [34] established the decay estimate by energy estimates, his approach does not seem to be available in a general Lp−Lq{L}^{p}-{L}^{q}setting. To overcome this difficulty, we study the decay property of an analytic C0{C}_{0}-semigroup associated with the linearized system. On the basis of a spectral analysis of the corresponding linear operator Aq{{\mathcal{A}}}_{q}defined in X0=Jq(F)×Bq,q2−1/q(G){X}_{0}={J}_{q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), we will show that −A˜q≔−Aq∣X˜0-{\widetilde{{\mathcal{A}}}}_{q}:= -{{\mathcal{A}}}_{q}{| }_{{\widetilde{X}}_{0}}generates an analytic C0{C}_{0}-semigroup {e−A˜qt}t≥0{\left\{{e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}\right\}}_{t\ge 0}in X˜0{\widetilde{X}}_{0}, which is exponentially stable on some subspace X˜0{\widetilde{X}}_{0}of X0{X}_{0}, see Section 5. Here, Jq(F){J}_{q}\left({\mathscr{F}})and Bq,q2−1/q(G){B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})stand for the solenoidal space of (Lq(F))3{\left({L}^{q}\left({\mathscr{F}}))}^{3}and the inhomogeneous Besov spaces, respectively, and the space X˜0{\widetilde{X}}_{0}is defined as the set of all (f,g)∈X0(f,g)\in {X}_{0}that satisfies the following orthogonal conditions: ∫Ffdy=∫Ff⋅(e3×y)dy=0,∫Ff⋅(eα×y)dy=ω∫Ggyαy3dG,(α=1,2),∫GgdG=∫GgyℓdG=0,(ℓ=1,2,3),\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f{\rm{d}}y& =& \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{3}\times y){\rm{d}}y=0,\\ \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{\alpha }\times y){\rm{d}}y& =& \omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}g{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},& \left(\alpha =1,2),\\ \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}g{\rm{d}}{\mathscr{G}}& =& \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}g{y}_{\ell }{\rm{d}}{\mathscr{G}}=0,& \left(\ell =1,2,3),\end{array}which are similar to but essentially different from the orthogonal condition considered in Shibata [26]. Thanks to these orthogonal conditions, we will observe that the resolvent set of −A˜q-{\widetilde{{\mathcal{A}}}}_{q}contains the right half-plane C+≔{λ∈C:Reλ≥0}{{\mathbb{C}}}_{+}:= \left\{\lambda \in {\mathbb{C}}\hspace{0.33em}:\hspace{0.33em}{\rm{Re}}\hspace{0.33em}\lambda \ge 0\right\}including λ=0\lambda =0, which implies the exponential stability of the semigroup. Hence, our discussion generalizes the argument employed in Shibata [26] who deals with the case ω=0\omega =0, and see also Shibata and Shimizu [28] for the case σ=ω=0\sigma =\omega =0. Notice that, if γ=0\gamma =0(i.e., ω=0\omega =0), then the equilibrium surface becomes a sphere so that a natural choice of G{\mathscr{G}}is a sphere as well. If we choose G{\mathscr{G}}as a sphere, we obtain a nice spectral property of the Laplace-Beltrami operator defined on G{\mathscr{G}}, which arises from the surface tension, see [26, Lemm. 4.5] (cf. [22, Prop. 10.2.1]). In our case, however, it follows from ω≠0\omega \ne 0and (1.2) that the equilibrium surface is not sphere. In addition, since we do not impose the smallness condition on ∣γ∣| \gamma | (as well as ∣ω∣| \omega | ), the equilibrium surface cannot be understood as a normal perturbation of a sphere in general, which means that Shibata’s approach fails. If we assume that ∣γ∣| \gamma | is sufficiently small, we can recover Shibata’s argument because the equilibrium surface is given by a normal perturbation of a sphere, and see a recent contribution by the author [40]. Thus, we have to introduce another technique to handle the term arising from the surface tension. To this end, we introduce the quadratic form that determines from the functional Eω{E}_{\omega }given in (1.7).It should be noted that our result even in the case γ=ω=0\gamma =\omega =0refines Shibata’s result [26]. In fact, in view of the trace theory (cf. Denk et al. [9]), if we study the linearized problem, the boundary data have to be in the intersection space:(1.8)Fp,qs(0,T;Lq(G))∩Lp(0,T;Bq,qs/2(G)),0<s<1,{F}_{p,q}^{s}\left(0,T;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}^{p}\left(0,T;\hspace{0.33em}{B}_{q,q}^{s\hspace{0.1em}\text{/}\hspace{0.1em}2}\left({\mathscr{G}})),\hspace{1.0em}0\lt s\lt 1,where Fp,qs{F}_{p,q}^{s}denotes the vector-valued inhomogeneous Triebel-Lizorkin spaces. Hence, to find the strong solution to (1.1) via a contraction mapping principle, we have to estimate the nonlinear terms in this intersection space. However, in the argument in Shibata [26], the boundary data were not lying in (1.8), and hence, his result is not optimal in view of trace theory. Besides, Shibata [26] did not investigate that the height function hhadmits the higher regularity Fp,q2−1/q(J;Lq(G)){F}_{p,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}})), which implies that the free interface can be understood in the classical sense due to the embedding Fp,q2−1/q(0,T;Lq(G))∩H1,p(0,T;Bq,q2−1/q(G))∩Lp(0,T;Bq,q3−1/q(G))↪C([0,T];C2(G))∩C1([0,T];C1(G)).{F}_{p,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(0,T;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {H}^{1,p}\left(0,T;\hspace{0.33em}{B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\cap {L}^{p}\left(0,T;\hspace{0.33em}{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\hspace{0.33em}\hookrightarrow \hspace{0.33em}C\left(\left[0,T];\hspace{0.33em}{C}^{2}\left({\mathscr{G}}))\cap {C}^{1}\left(\left[0,T];\hspace{0.33em}{C}^{1}\left({\mathscr{G}})).To overcome these fallacious, we rely on the recent contributions established by the author [39,40], which were based on the studies of Lindemulder [13] and Meyries and Veraar [14,15]. Furthermore, it was not showed in the study by Shibata [25,26] that the solution can be understood in the classical sense, but the well-known parameter trick (cf. Prüss-Simonett [22, Ch. 9]) implies that the solution is indeed real analytic, jointly in time and space. This shows that the solutions to (1.1) are indeed classical.To explain our main result, we shall introduce the quadratic form that characterize the stability of the stationary solution to (1.1). As we will explain in the next section, the free surface Γ(t)\Gamma \left(t)can be approximated by a real analytic surface G{\mathscr{G}}in the sense that the Hausdorff distance of the second-order bundles of Γ(t)\Gamma \left(t)and G{\mathscr{G}}is as small as we wish. In this case, we can write (1.9)Γ(t)={p+h(p,t)νG(p):p∈G},Γ0={p+h0(p)νG(p):p∈G},\Gamma \left(t)=\left\{p+h\left(p,t){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {\mathscr{G}}\right\},\hspace{1em}{\Gamma }_{0}=\left\{p+{h}_{0}\left(p){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {\mathscr{G}}\right\},where hhis an unknown function but h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})is a given function. Here, we will prove that hhis indeed smooth function defined on G{\mathscr{G}}for each t>0t\gt 0. Since we seek global solutions that converges to an equilibrium, in the following, we may set G=Γ∞{\mathscr{G}}={\Gamma }_{\infty }. First, we observe that (1.2) is the Euler-Lagrange equation associated with Eω{E}_{\omega }. In fact, the first variation of Eω{E}_{\omega }at h=0h=0is given by (1.10)δ0Eω=−∫GσHGhdΓ−∫Gω22∣y′∣2hdG−∫Gp0hdG.{\delta }_{0}{E}_{\omega }=-\mathop{\int }\limits_{{\mathscr{G}}}\sigma {{\mathscr{H}}}_{{\mathscr{G}}}h{\rm{d}}\Gamma -\mathop{\int }\limits_{{\mathscr{G}}}\frac{{\omega }^{2}}{2}| y^{\prime} {| }^{2}h{\rm{d}}{\mathscr{G}}-\mathop{\int }\limits_{{\mathscr{G}}}{p}_{0}h{\rm{d}}{\mathscr{G}}.In addition, the second variation of Eω{E}_{\omega }at h=0h=0is given by (1.11)δ02Eω=−∫Gσ(hΔGh−2KGh2)dG−∫Gω22∂∂νG∣y′∣2−∣y′∣2HGh2dG+∫Gp0HGh2dG,{\delta }_{0}^{2}{E}_{\omega }=-\mathop{\int }\limits_{{\mathscr{G}}}\sigma (h{\Delta }_{{\mathscr{G}}}h-2{{\mathscr{K}}}_{{\mathscr{G}}}{h}^{2}){\rm{d}}{\mathscr{G}}-\mathop{\int }\limits_{{\mathscr{G}}}\frac{{\omega }^{2}}{2}\left(\frac{\partial }{\partial {\nu }_{{\mathscr{G}}}}| y^{\prime} {| }^{2}-| y^{\prime} {| }^{2}{{\mathscr{H}}}_{{\mathscr{G}}}\right){h}^{2}{\rm{d}}{\mathscr{G}}+\mathop{\int }\limits_{{\mathscr{G}}}{p}_{0}{{\mathscr{H}}}_{{\mathscr{G}}}{h}^{2}{\rm{d}}{\mathscr{G}},where KG{{\mathscr{K}}}_{{\mathscr{G}}}is the Gaussian curvature. We refer to [32, Sec. 2] for the derivations of (1.10) and (1.11), and see also [22, Ch. 2] for further geometric background. Recalling (1.2), we have σHG2+ω22∣y′∣2HG+p0HG=0onG,\sigma {{\mathscr{H}}}_{{\mathscr{G}}}^{2}+\frac{{\omega }^{2}}{2}| y^{\prime} {| }^{2}{{\mathscr{H}}}_{{\mathscr{G}}}+{p}_{0}{{\mathscr{H}}}_{{\mathscr{G}}}=0\hspace{1.0em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}},and hence, the second variation δ02Eω{\delta }_{0}^{2}{E}_{\omega }of Eω{E}_{\omega }can be rewritten as follows: δ02Eω=−∫GσhHG′(0)hdG−∫Gω22∂∂νG∣y′∣2h2dG≕∫GhℬGhdG,{\delta }_{0}^{2}{E}_{\omega }=-\mathop{\int }\limits_{{\mathscr{G}}}\sigma h{{\mathscr{H}}}_{{\mathscr{G}}}^{^{\prime} }\left(0)h{\rm{d}}{\mathscr{G}}-\mathop{\int }\limits_{{\mathscr{G}}}\frac{{\omega }^{2}}{2}\frac{\partial }{\partial {\nu }_{{\mathscr{G}}}}| y^{\prime} {| }^{2}{h}^{2}{\rm{d}}{\mathscr{G}}\hspace{0.33em}=: \hspace{0.33em}\mathop{\int }\limits_{{\mathscr{G}}}h{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}},where HG′(0){{\mathscr{H}}}_{{\mathscr{G}}}^{^{\prime} }\left(0)stands for the first variation of HG{{\mathscr{H}}}_{{\mathscr{G}}}at h=0h=0, since it holds ΔGh−2KGh=HG′(0)h−HG2h.{\Delta }_{{\mathscr{G}}}h-2{{\mathscr{K}}}_{{\mathscr{G}}}h={{\mathscr{H}}}_{{\mathscr{G}}}^{^{\prime} }\left(0)h-{{\mathscr{H}}}_{{\mathscr{G}}}^{2}h.Since δ0Eω=0{\delta }_{0}{E}_{\omega }=0, we observe that G{\mathscr{G}}is a minimal surface. We now normalize ℬGh{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}hby ℬ^Gh=ℬGh−1∣G∣∫GℬGhdG{\widehat{{\mathcal{ {\mathcal B} }}}}_{{\mathscr{G}}}h={{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h-\frac{1}{| {\mathscr{G}}| }\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}and define the quadratic form ΨG{\Psi }_{{\mathscr{G}}}by (1.12)ΨG(g,h)≔∫Ggℬ^GhdG{\Psi }_{{\mathscr{G}}}\left(g,h):= \mathop{\int }\limits_{{\mathscr{G}}}g{\widehat{{\mathcal{ {\mathcal B} }}}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}for g,h∈H2,2(G)g,h\in {H}^{2,2}\left({\mathscr{G}}). Then, we have δ02Eω=ΨG(h,h){\delta }_{0}^{2}{E}_{\omega }={\Psi }_{{\mathscr{G}}}\left(h,h)and ∫Gℬ^GhdG=0{\int }_{{\mathscr{G}}}{\widehat{{\mathcal{ {\mathcal B} }}}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}=0.The aim of this article is to show the stability of the stationary solution (v∞,π∞,G)\left({v}_{\infty },{\pi }_{\infty },{\mathscr{G}})to (1.1), provided that there exists a solution G{\mathscr{G}}to (1.2), and the quadratic form ΨG{\Psi }_{{\mathscr{G}}}satisfies the following assumption.Assumption 1.1The quadratic form ΨG(h,h){\Psi }_{{\mathscr{G}}}\left(h,h)is positive definite on L(0)2(G)≔h∈L2(G):∫GhdG=∫GhyjdG=0,j=1,2,3.,{L}_{\left(0)}^{2}\left({\mathscr{G}}):= \left\{h\in {L}^{2}\left({\mathscr{G}})\hspace{0.33em}:\hspace{0.33em}\mathop{\int }\limits_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}=\mathop{\int }\limits_{{\mathscr{G}}}h{y}_{j}{\rm{d}}{\mathscr{G}}=0,\hspace{1.0em}j=1,2,3.\right\},that is, there exists a constant ccsuch that ∣ΨG(h,h)∣L2(G)≥c∣h∣L2(G)2| {\Psi }_{{\mathscr{G}}}\left(h,h){| }_{{L}^{2}\left({\mathscr{G}})}\ge c| h{| }_{{L}^{2}\left({\mathscr{G}})}^{2}holds for every h∈L(0)2(G)∩H2,2(G)h\in {L}_{\left(0)}^{2}\left({\mathscr{G}})\hspace{0.33em}\cap \hspace{0.33em}{H}^{2,2}\left({\mathscr{G}}).Our main result reads as follows.Theorem 1.2Suppose that Ω(0)\Omega \left(0)satisfies (1.4)3{\left(1.4)}_{3}. Assume that there exists a smooth solution Γ∞{\Gamma }_{\infty }to (1.2), which is rotationally symmetric about the x3{x}_{3}axis and that the quadratic form ΨG{\Psi }_{{\mathscr{G}}}satisfies Assumption 1.1. Let (v∞,π∞)\left({v}_{\infty },{\pi }_{\infty })be given by (1.3). If pp, qq, and δ\delta satisfy(1.13)2<p<∞,3<q<∞,1p+32q<δ−12≤12,2\lt p\lt \infty ,\hspace{1.0em}3\lt q\lt \infty ,\hspace{1.0em}\frac{1}{p}+\frac{3}{2q}\lt \delta -\frac{1}{2}\le \frac{1}{2},then an equilibrium (v∞,π∞,Γ∞)\left({v}_{\infty },{\pi }_{\infty },{\Gamma }_{\infty })is stable in the following sense: Let Γ0{\Gamma }_{0}be given by (1.9)2{\left(1.9)}_{2}with a given function h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}). There exists a positive constant ε>0\varepsilon \gt 0such that for all v0−v∞∈Bq,p2(δ−1/p)(Ω(0)){v}_{0}-{v}_{\infty }\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(0))and h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfying the smallness condition: ∣v0−v∞∣Bq,p2(δ−1/p)(Ω(0))+∣h0∣Bq,p2+δ−1/p−1/q(G)≤ε,| {v}_{0}-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(0))}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}\le \varepsilon ,the compatibility conditions: (1.14)divv0=0inΩ,PΓ0[μ(∇v0+[∇v0]⊤)]=0onΓ0{\rm{div}}\hspace{0.33em}{v}_{0}=0\hspace{1.0em}{in}\hspace{0.33em}\Omega ,\hspace{1.0em}{{\mathcal{P}}}_{{\Gamma }_{0}}\left[\mu \left(\nabla {v}_{0}+{\left[\nabla {v}_{0}]}^{\top })]=0\hspace{1.0em}{on}\hspace{0.33em}{\Gamma }_{0}and the conditions ∫Ω(0)v0(x)dx=0{\int }_{\Omega \left(0)}{v}_{0}\left(x){\rm{d}}x=0and ∫Ω(0)(v0×x)dx=γe3{\int }_{\Omega \left(0)}\left({v}_{0}\times x){\rm{d}}x=\gamma {e}_{3}with a constant γ∈R\gamma \in {\mathbb{R}}, there exists a unique global classical solution (v(t),π(t),Γ(t))\left(v\left(t),\pi \left(t),\Gamma \left(t))of problem (1.1). In particular, the set ⋃t∈(0,∞)(Γ(t)×{t}){\bigcup }_{t\in \left(0,\infty )}\left(\Gamma \left(t)\times \left\{t\right\})is real analytic manifold and the function (v,π):{(x,t)∈Ω(t)×(0,∞)}→R4\left(v,\pi ):\left\{\left(x,t)\in \Omega \left(t)\times \left(0,\infty )\right\}\to {{\mathbb{R}}}^{4}is real analytic. In addition, if Γ(t)\Gamma \left(t)is parameterized over G{\mathscr{G}}by means of a height function h(t)h\left(t), that is, if Γ(t)\Gamma \left(t)is given by (1.9)1{\left(1.9)}_{1}with an unknown function h, it holds∣v(t)−v∞∣Bq,p2(δ−1/p)(Ω(t))=O(e−ct)and∣h(t)∣Bq,p2+δ−1/p−1/q(G)=O(e−ct)ast→∞| v\left(t)-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(t))}=O\left({e}^{-ct})\hspace{1.0em}{and}\hspace{1.0em}| h\left(t){| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}=O\left({e}^{-ct})\hspace{1.0em}{as}\hspace{0.33em}t\to \infty with some positive constant c. Here, we have set PΓ0≔I−νΓ0⊗νΓ0{{\mathcal{P}}}_{{\Gamma }_{0}}:= I-{\nu }_{{\Gamma }_{0}}\otimes {\nu }_{{\Gamma }_{0}}.Remark 1.3The restriction (1.13) implies the embeddings Bq,p2(δ−1/p)(Ω(t))↪BUC1(Ω(t))andBq,p2+δ−1/p−1/q(G)↪BUC2(G).{B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(t))\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{1}\left(\Omega \left(t))\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{2}\left({\mathscr{G}}).Hence, we clearly observe that ∣v(t)−v∞∣C1(Ω(t))=O(e−ct)| v\left(t)-{v}_{\infty }{| }_{{C}^{1}\left(\Omega \left(t))}=O\left({e}^{-ct})and ∣h(t)∣C2(G)=O(e−ct)| h\left(t){| }_{{C}^{2}\left({\mathscr{G}})}=O\left({e}^{-ct})as t→∞t\to \infty . We also notice that the well known norm equivalence of the Hölder spaces Cs≃B∞,∞s{C}^{s}\simeq {B}_{\infty ,\infty }^{s}(s>0,s∉Ns\gt 0,s\notin {\mathbb{N}}) implies the embedding properties C2+α(Ω(0))≃B∞,∞2+α(Ω(0))↪Bq,p2(δ−1/p)(Ω(0)),α∈(0,1),C3+α(G)≃B∞,∞3+α(G)↪Bq,p2+δ−1/p−1/q(G),α∈(0,1),\begin{array}{rcl}{C}^{2+\alpha }\left(\Omega \left(0))& \simeq & {B}_{\infty ,\infty }^{2+\alpha }\left(\Omega \left(0))\hspace{0.33em}\hookrightarrow \hspace{0.33em}{B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(0)),\hspace{1.0em}\alpha \in \left(0,1),\\ {C}^{3+\alpha }\left({\mathscr{G}})& \simeq & {B}_{\infty ,\infty }^{3+\alpha }\left({\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}),\hspace{1.0em}\alpha \in \left(0,1),\end{array}which shows that the regularity of v0−v∞{v}_{0}-{v}_{\infty }and h0{h}_{0}in Theorem 1.2 are less than the assumption imposed in [34, Thm. 2.1]. Hence, Theorem 1.2 indeed improves the result due to Solonnikov [34, Thm. 2.1]. Furthermore, in contrast to [34, Thm. 2.1], solutions regularize and immediately become real analytic in space and time, which encodes typical parabolic behavior.Remark 1.4If the initial velocity v0{v}_{0}satisfies the orthogonal condition (1.4) with γ=0\gamma =0, Theorem 1.2 can be regarded as a generalization of Shibata [26].Remark 1.5Using elliptic integrals, the existence of the equilibrium surface Γ∞{\Gamma }_{\infty }that satisfies (1.6) may be shown and, especially, it is simply connected if the value of the nondimensional parameter h=ω2a38σ{\mathfrak{h}}=\frac{{\omega }^{2}{a}^{3}}{8\sigma }is strictly less than some value hmax≈2.32911{{\mathfrak{h}}}_{{\rm{\max }}}\approx 2.32911, and see Chandrasekhar [7] (cf. Appell [2, Ch. IX]). Here, aais the equatorial radius of the liquid drop and Γ∞{\Gamma }_{\infty }is symmetric with respect to an x1{x}_{1}-x2{x}_{2}plane. In the following, let us briefly explain the result presented by Chandrasekhar [7]. If h=0{\mathfrak{h}}=0(i.e., ω=0\omega =0), it is clear that Γ∞{\Gamma }_{\infty }is the sphere with the radius R>0R\gt 0; if 0<h≤10\lt {\mathfrak{h}}\le 1, then Γ∞{\Gamma }_{\infty }becomes an ellipsoid; if 1<h<hmax1\lt {\mathfrak{h}}\lt {{\mathfrak{h}}}_{{\rm{\max }}}, then a dimple appears. See also [7, Fig. 1] for a precise information. On the other hand, by [2, Ch. IX], if h≥hmax{\mathfrak{h}}\ge {{\mathfrak{h}}}_{{\rm{\max }}}, the equilibrium surface Γ∞{\Gamma }_{\infty }becomes a toroid (i.e., not simply connected), which replicates the classical experiment due to Plateau in the middle of the 19th century. In this article, it is not necessary to assume that the free boundary Γ(t)\Gamma \left(t)is simply connected, which is the same as Solonnikov [32], since we do not need to use polar coordinates to express the free boundary like Padula and Solonnikov [19]. Notice that Theorem 1.2 justifies the stability result, obtained by Brown and Scriven [6], for an axisymmetric equilibrium figure in the sense that we also consider the motion of the incompressible viscous fluid occupied inside the free boundary.In contrast to the previous article [40], Theorem 1.2 replaces the smallness condition on ∣γ∣| \gamma | by the condition of the positivity of δ02Eω=ΨG(h,h){\delta }_{0}^{2}{E}_{\omega }={\Psi }_{{\mathscr{G}}}\left(h,h), where it is assumed the existence of the equilibrium surface Γ∞{\Gamma }_{\infty }satisfying (1.6). However, Theorem 1.2 does not conflict with the result obtained in [40]. In fact, if ∣γ∣| \gamma | is suitably small, it follows from [40, Prop A.1] that there exists a unique Γ∞{\Gamma }_{\infty }satisfying (1.2) and (1.6) with Γ∞={p+h∞(p)νG(p):p∈SR},{\Gamma }_{\infty }=\left\{p+{h}_{\infty }\left(p){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {S}_{R}\right\},where ∣(h∞,∇h∞)∣L∞(SR)| \left({h}_{\infty },\nabla {h}_{\infty }){| }_{{L}^{\infty }\left({S}_{R})}are small. Here, SR{S}_{R}is a sphere centered at the origin with a radius R>0R\gt 0, and ∣(h∞,∇h∞)∣L∞(SR)| \left({h}_{\infty },\nabla {h}_{\infty }){| }_{{L}^{\infty }\left({S}_{R})}can be dominated by ∣γ∣| \gamma | , see [40, Prop. A.1]. In this case, setting p0=2σ/R{p}_{0}=2\sigma \hspace{0.1em}\text{/}\hspace{0.1em}R, the functional δ02Eω{\delta }_{0}^{2}{E}_{\omega }can be regarded as a perturbation from −∫SRσ(h¯ΔSRh¯−2KSRh¯2)dSR(h¯≔h−h∞),-\mathop{\int }\limits_{{S}_{R}}\sigma (\overline{h}{\Delta }_{{S}_{R}}\overline{h}-2{{\mathscr{K}}}_{{S}_{R}}{\overline{h}}^{2}){\rm{d}}{S}_{R}\hspace{1.0em}\left(\overline{h}:= h-{h}_{\infty }),which is positive definite on L(0)2(SR)∩H2,2(SR){L}_{\left(0)}^{2}\left({S}_{R})\cap {H}^{2,2}\left({S}_{R})(see [22, Prop. 10.2.1]). Hence, taking ∣γ∣| \gamma | as small as possible, we observe that the smallness of ∣γ∣| \gamma | implies the positivity of δ02Eω{\delta }_{0}^{2}{E}_{\omega }. Thus, we can conclude that Theorem 1.2 is an extension of the result obtained in the previous article. Accordingly, Theorem 1.2 includes all results obtained in [26,34,40].The rest of the article is folded as follows. In Section 2, we recall the notation of functional spaces and preliminary propositions used throughout this article. In Section 3, we transform the system (1.1) to a problem on a domain F{\mathscr{F}}surrounded by a fixed interface G{\mathscr{G}}in terms of the Hanzawa transform. Section 4 is devoted to showing that the principal part of the linearization has the property of maximal Lp−Lq{L}^{p}-{L}^{q}regularity. Then, some exponential decay property of the linearized system is proved in Section 5. Finally, the Section 6 presents the proof of the main result, Theorem 1.2.2Preliminaries2.1NotationsLet us fix the notations in this article. Let N{\mathbb{N}}be the set of all natural numbers and N0≔N∪{0}{{\mathbb{N}}}_{0}:= {\mathbb{N}}\cup \left\{0\right\}, and let R{\mathbb{R}}and C{\mathbb{C}}be, respectively, the set of all real numbers and the set of all complex numbers. Set R+≔{a∈R:a>0}{{\mathbb{R}}}_{+}:= \left\{a\in {\mathbb{R}}\hspace{0.33em}:\hspace{0.33em}a\gt 0\right\}. By C>0C\gt 0, we will often denote a generic constant that does not depend on the quantities at stake.2.2Functional spacesIn this subsection, we introduce functional spaces used throughout this article. Let p,q∈[1,∞]p,q\in \left[1,\infty ]. For any DDdomain of R3{{\mathbb{R}}}^{3}, we denote the standard K{\mathbb{K}}-valued Lebesgue spaces and Sobolev spaces on DDby Lq(D){L}^{q}\left(D)and Hm,q(D){H}^{m,q}\left(D), m∈Nm\in {\mathbb{N}}, respectively, where K∈{R,C}{\mathbb{K}}\in \left\{{\mathbb{R}},{\mathbb{C}}\right\}. The standard K{\mathbb{K}}-valued inhomogeneous Besov spaces on DDare denoted by Bp,qs(D){B}_{p,q}^{s}\left(D), s∈Rs\in {\mathbb{R}}. The homogeneous spaces H˙1,q(D){\dot{H}}^{1,q}\left(D)are given by H˙1,q(D)≔{w∈Lloc1(D):∇w∈Lq(D)}{\dot{H}}^{1,q}\left(D):= \left\{w\in {L}_{{\rm{loc}}}^{1}\left(D)\hspace{0.33em}:\hspace{0.33em}\nabla w\in {L}^{q}\left(D)\right\}for q∈(1,∞)q\in \left(1,\infty )and a domain D⊂R3D\subset {{\mathbb{R}}}^{3}, while the dual space of H˙1,q(D){\dot{H}}^{1,q}\left(D)is written by H˙−1,q′(D){\dot{H}}^{-1,q^{\prime} }\left(D)with the Hölder conjugate q′≔q/(q−1)q^{\prime} := q\hspace{0.1em}\text{/}\hspace{0.1em}\left(q-1)of qq.Let XXbe a Banach space. Then the mm-product space of XXis denoted by Xm{X}^{m}, while its norm is usually denoted by ∣⋅∣X| \cdot {| }_{X}instead of ∣⋅∣Xm| \cdot {| }_{{X}^{m}}when no confusion can arise.Let Banach spaces X1{X}_{1}and X2{X}_{2}the norm of X1∩X2{X}_{1}\cap {X}_{2}is denoted by ∣⋅∣X1∩X2≔∣⋅∣X1+∣⋅∣X2| \cdot {| }_{{X}_{1}\cap {X}_{2}}:= | \cdot {| }_{{X}_{1}}+| \cdot {| }_{{X}_{2}}, and ℒ(X1,X2){\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{2})stands for the Banach space of all bounded linear operators from X1{X}_{1}to X2{X}_{2}. We may write ℒ(X1){\mathcal{ {\mathcal L} }}\left({X}_{1})instead of ℒ(X1,X1){\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{1})to shorten the notation. The symbol Hol(Λ,ℒ(X1,X2)){\rm{Hol}}\hspace{0.33em}\left(\Lambda ,{\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{2}))represents the set of all ℒ(X1,X2){\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{2})-valued holomorphic functions defined on Λ⊂C\Lambda \subset {\mathbb{C}}. We set Σθ,z0≔{z∈C⧹{0}:∣argz∣≤π−θ,∣z∣≥z0}{\Sigma }_{\theta ,{z}_{0}}:= \left\{z\in {\mathbb{C}}\setminus \left\{0\right\}\hspace{0.33em}:\hspace{0.33em}| \arg z| \le \pi -\theta ,| z| \ge {z}_{0}\right\}with θ∈(0,π/2)\theta \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2)and z0>0{z}_{0}\gt 0.For I⊂RI\subset {\mathbb{R}}and p∈(1,∞]p\in \left(1,\infty ], let Lp(I;X){L}^{p}\left(I;\hspace{0.33em}X)and H1,p(I;X){H}^{1,p}\left(I;\hspace{0.33em}X)be the XX-valued Lebesgue spaces on IIand the XX-valued Sobolev spaces on II, respectively. For p∈(1,∞)p\in \left(1,\infty )and δ∈(1/p,1]\delta \in \left(1\hspace{0.1em}\text{/}\hspace{0.1em}p,1], we set Lδp(I;X)≔{f:I→X:t1−δf∈Lp(I;X)},Hδ1,p(I;X)≔{f∈Lδp(I;X)∩H1,1(I;X):∂tf∈Lδp(I;X)}.\begin{array}{rcl}{L}_{\delta }^{p}\left(I;\hspace{0.33em}X)& := & \{f:I\to X\hspace{0.33em}:\hspace{0.33em}{t}^{1-\delta }f\in {L}^{p}\left(I;\hspace{0.33em}X)\},\\ {H}_{\delta }^{1,p}\left(I;\hspace{0.33em}X)& := & \{f\in {L}_{\delta }^{p}\left(I;\hspace{0.33em}X)\cap {H}^{1,1}\left(I;\hspace{0.33em}X)\hspace{0.33em}:\hspace{0.33em}{\partial }_{t}f\in {L}_{\delta }^{p}\left(I;\hspace{0.33em}X)\}.\end{array}For p∈(1,∞)p\in \left(1,\infty ), q∈[1,∞]q\in \left[1,\infty ], and s∈Rs\in {\mathbb{R}}, the symbol Fp,q,δs(I;X){F}_{p,q,\delta }^{s}\left(I;\hspace{0.33em}X)stands for the XX-valued Triebel-Lizorkin spaces with the power weight ∣t∣p(1−δ)| t{| }^{p\left(1-\delta )}. In addition, the Banach space of all XX-valued bounded uniformly continuous functions on IIis denoted by BUC(I;X){\rm{BUC}}\left(I;\hspace{0.33em}X). Finally, BUCm(I;X){{\rm{BUC}}}^{m}\left(I;\hspace{0.33em}X)is the subset of BUC(I;X){\rm{BUC}}\left(I;\hspace{0.33em}X)that has bounded partial derivatives up to order m∈Nm\in {\mathbb{N}}. Here, BUC(D){\rm{BUC}}\left(D)and BUCm(D){{\rm{BUC}}}^{m}\left(D)are defined similarly as mentioned earlier. For further information on function spaces, we refer the reader to [22,38].2.3The space of data for divergence equationAs we will see in Section 2.4, in a transformed system, we have the divergence equation divu=gd{\rm{div}}\hspace{0.33em}u={g}_{d}in F{\mathscr{F}}. Hence, to deal with this equation, it is required to introduce a space DIq(F){{\rm{DI}}}_{q}\left({\mathscr{F}})that is the set of all gd∈H1,q(F){g}_{d}\in {H}^{1,q}\left({\mathscr{F}})such that there exists a solution g˜d∈Lq(F)3{\widetilde{g}}_{d}\in {L}^{q}{\left({\mathscr{F}})}^{3}to (gd,φ)F=−(g˜d,∇φ)F{\left({g}_{d},\varphi )}_{{\mathscr{F}}}=-{\left({\widetilde{g}}_{d},\nabla \varphi )}_{{\mathscr{F}}}for every φ∈H01,q′(F)≔{φ∈H1,q(F):φ∣G=0}\varphi \in {H}_{0}^{1,q^{\prime} }\left({\mathscr{F}}):= \left\{\varphi \in {H}^{1,q}\left({\mathscr{F}})\hspace{0.33em}:\hspace{0.33em}\varphi {| }_{{\mathscr{G}}}=0\right\}. Here and in the following, (⋅,⋅)F{\left(\cdot ,\cdot )}_{{\mathscr{F}}}denotes the L2{L}^{2}inner product in F{\mathscr{F}}. We now define G(gd)≔{g∗∈Lq(F)3:divg˜d=divg∗}{\mathsf{G}}\left({g}_{d}):= \left\{{g}_{\ast }\in {L}^{q}{\left({\mathscr{F}})}^{3}\hspace{0.33em}:\hspace{0.33em}{\rm{div}}\hspace{0.33em}{\widetilde{g}}_{d}={\rm{div}}\hspace{0.33em}{g}_{\ast }\right\}and denote the representative elements of G(gd){\mathsf{G}}\left({g}_{d})by [G(gd)]\left[{\mathsf{G}}\left({g}_{d})]. For brevity, and when there is no danger of confusion, we write G(gd){\mathsf{G}}\left({g}_{d})instead of [G(gd)]\left[{\mathsf{G}}\left({g}_{d})]. Here, it holds divG(gd)=gd{\rm{div}}\hspace{0.33em}{\mathsf{G}}\left({g}_{d})={g}_{d}in F{\mathscr{F}}for every gd∈DIq(F){g}_{d}\in {{\rm{DI}}}_{q}\left({\mathscr{F}}). Setting ∣gd∣DIq(F)≔∣gd∣H1,q(F)+infg∗∈G(gd)∣g∗∣Lq(F)forgd∈DIq(F),| {g}_{d}{| }_{{{\rm{DI}}}_{q}\left({\mathscr{F}})}:= | {g}_{d}{| }_{{H}^{1,q}\left({\mathscr{F}})}+\mathop{\inf }\limits_{{g}_{\ast }\in {\mathsf{G}}\left({g}_{d})}| {g}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}{g}_{d}\in {{\rm{DI}}}_{q}\left({\mathscr{F}}),we observe that DIq(F){{\rm{DI}}}_{q}\left({\mathscr{F}})is a Banach space endowed with the norm ∣⋅∣DIq(F)| \cdot {| }_{{{\rm{DI}}}_{q}\left({\mathscr{F}})}.2.4ℛ{\mathcal{ {\mathcal R} }}-bounded families of operatorsWe next recall the basic theory of the ℛ{\mathcal{ {\mathcal R} }}-boundedness of a family of operators, which will be used in Section 4. Here, we refer to [8] for the fundamental concept of ℛ{\mathcal{ {\mathcal R} }}-boundedness. In the following, the ℛ{\mathcal{ {\mathcal R} }}-bound of a family of operators T⊂ℒ(X,Y){\mathcal{T}}\subset {\mathcal{ {\mathcal L} }}\left(X,Y)is denoted by ℛX→Y{T}{{\mathcal{ {\mathcal R} }}}_{X\to Y}\left\{{\mathcal{T}}\right\}, where XXand YYare Banach spaces. If X=YX=Y, we ofren write ℛX{T}{{\mathcal{ {\mathcal R} }}}_{X}\left\{{\mathcal{T}}\right\}for short. The following result is a direct consequence of [8, Rem. 3.2 (4)].Proposition 2.1Let 1≤q<∞1\le q\lt \infty and G⊂R3G\subset {{\mathbb{R}}}^{3}be a domain. Let m(λ)m\left(\lambda )be a bounded function defined on a subset Λ\Lambda of C{\mathbb{C}}, and let Mm(λ){M}_{m}\left(\lambda )be a multiplication operator given by Mm(λ)f≔m(λ)f{M}_{m}\left(\lambda )f:= m\left(\lambda )ffor every f∈Lq(G)f\in {L}^{q}\left(G\right). Then it holdsℛLq(G)({Mm(λ):λ∈Λ})≤Kq2(supλ∈Λ∣m(λ)∣),{{\mathcal{ {\mathcal R} }}}_{{L}^{q}\left(G)}\left(\left\{{M}_{m}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in \Lambda \right\})\le {K}_{q}^{2}\left(\mathop{\sup }\limits_{\lambda \in \Lambda }| m\left(\lambda \right)| \right),where Kq>0{K}_{q}\gt 0is a constant appearing in the Khintchine inequality.3Reduction to a fixed reference surfaceIn this section, we transform problem (1.1) to one on the fixed spatial domain t≥0t\ge 0. Putting V≔v−v∞V:= v-{v}_{\infty }and P=π−π∞P=\pi -{\pi }_{\infty }, the problem of a stability of the equilibrium (v∞,π∞,Γ∞)\left({v}_{\infty },{\pi }_{\infty },{\Gamma }_{\infty })reduces to a free boundary problem for the perturbation (V,P)\left(V,P): (3.1)∂tV+(v∞⋅∇)V+(V⋅∇)v∞+(V⋅∇)V−μΔV+∇P=0,inΩ(t),divV=0,inΩ(t),S(V,P)νΓ=σHΓ+ω22∣x′∣2+p0νΓ,onΓ(t),VΓ=(v∞+V)⋅νΓ,onΓ(t),V(0)=v0−v∞,inΩ(0),Γ(0)=Γ0.\left\{\begin{array}{ll}{\partial }_{t}V+\left({v}_{\infty }\cdot \nabla )V+\left(V\cdot \nabla ){v}_{\infty }+\left(V\cdot \nabla )V-\mu \Delta V+\nabla P=0,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \left(t),\\ {\rm{div}}\hspace{0.33em}V=0,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\hspace{0.33em}\Omega \left(t),\\ S\left(V,P){\nu }_{\Gamma }=\left(\sigma {{\mathscr{H}}}_{\Gamma }+\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{\Gamma },& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\Gamma \left(t),\\ {V}_{\Gamma }=\left({v}_{\infty }+V)\cdot {\nu }_{\Gamma },& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\Gamma \left(t),\\ V\left(0)={v}_{0}-{v}_{\infty },& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \left(0),\\ \Gamma \left(0)={\Gamma }_{0}.& \end{array}\right.In our analysis, it will be useful to eliminate the terms (v∞⋅∇)V\left({v}_{\infty }\cdot \nabla )Vand (V⋅∇)v∞\left(V\cdot \nabla ){v}_{\infty }. To this end, we introduce the coordinate system rotating about the x3{x}_{3}axis with a constant angular velocity ω∈R\omega \in {\mathbb{R}}, i.e., x=O(ωt)zx={\mathcal{O}}\left(\omega t)z. We now set V˜(z,t)≔O−1(ωt)V(O(ωt)z,t)andP˜(z,t)≔P(O(ωt)z,t)\widetilde{V}\left(z,t):= {{\mathcal{O}}}^{-1}\left(\omega t)V\left({\mathcal{O}}\left(\omega t)z,t)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\widetilde{P}\left(z,t):= P\left({\mathcal{O}}\left(\omega t)z,t)with O(θ)≔cosθ−sinθ0sinθcosθ0001.{\mathcal{O}}\left(\theta ):= \left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right).Then problem (3.1) can be rewritten as follows: (3.2)∂tV˜+(V˜⋅∇)V˜−μΔV˜+2ω(e3×V˜)+∇P˜=0,in Ω˜(t),divV˜=0,in Ω˜(t),S(V˜,P˜)νΓ˜=σHΓ˜+ω22∣z′∣2+p0νΓ˜,on Γ˜(t),VΓ˜=V˜⋅νΓ˜,on Γ˜(t),V˜(0)=v0−v∞≕V˜0,in Ω˜(0),Γ(0)=Γ0,\left\{\begin{array}{ll}{\partial }_{t}\widetilde{V}+\left(\widetilde{V}\cdot \nabla )\widetilde{V}-\mu \Delta \widetilde{V}+2\omega \left({e}_{3}\times \widetilde{V})+\nabla \widetilde{P}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\widetilde{\Omega }\left(t)\text{},\\ {\rm{div}}\hspace{0.33em}\widetilde{V}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\widetilde{\Omega }\hspace{0.33em}\left(t)\text{},\\ S\left(\widetilde{V},\widetilde{P}){\nu }_{\widetilde{\Gamma }}=\left(\sigma {{\mathscr{H}}}_{\widetilde{\Gamma }}+\frac{{\omega }^{2}}{2}| z^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{\widetilde{\Gamma }},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\widetilde{\Gamma }\hspace{0.33em}\left(t)\text{},\\ {V}_{\widetilde{\Gamma }}=\widetilde{V}\cdot {\nu }_{\widetilde{\Gamma }},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\widetilde{\Gamma }\hspace{0.33em}\left(t)\text{},\\ \widetilde{V}\left(0)={v}_{0}-{v}_{\infty }\hspace{0.33em}=: \hspace{0.33em}{\widetilde{V}}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\widetilde{\Omega }\left(0)\text{},\\ \Gamma \left(0)={\Gamma }_{0},& \end{array}\right.where we have set Ω˜(t)=O−1(ωt)Ω(t)\widetilde{\Omega }\left(t)={{\mathcal{O}}}^{-1}\left(\omega t)\Omega \left(t)and Γ˜(t)=∂Ω˜(t)\widetilde{\Gamma }\left(t)=\partial \widetilde{\Omega }\left(t). Here, VΓ˜(⋅,t){V}_{\widetilde{\Gamma }}\left(\cdot ,t)and HΓ˜(⋅,t){{\mathscr{H}}}_{\widetilde{\Gamma }}\left(\cdot ,t)stand for the normal velocity and the doubled mean curvature of Γ˜(t)\widetilde{\Gamma }\left(t)with respect to the norm νΓ˜(⋅,t)=O−1(ωt)νΓ(⋅,t){\nu }_{\widetilde{\Gamma }}\left(\cdot ,t)={{\mathcal{O}}}^{-1}\left(\omega t){\nu }_{\Gamma }\left(\cdot ,t)to Γ˜(t)\widetilde{\Gamma }\left(t), respectively. For further details of the derivation of (3.2), the readers may consult the discussion in [34]. It follows from conditions (1.4)3{\left(1.4)}_{3}and (1.5) that ∫Ω˜(t)zdz=∫Ω˜(0)dz=0,\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}z{\rm{d}}z=\mathop{\int }\limits_{\widetilde{\Omega }\left(0)}{\rm{d}}z=0,i.e., the barycenter of the fluid is still located at the origin. In addition, condition (1.4)1{\left(1.4)}_{1}becomes ∫Ω˜(t)V˜(z,t)dz=0\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}\left(z,t){\rm{d}}z=0due to ∫Ω˜(t)zdz=0{\int }_{\widetilde{\Omega }\left(t)}z{\rm{d}}z=0. Notice that (3.2)4{\left(3.2)}_{4}is equivalent to VΓ˜(z,t)=V˜⋅νΓ˜−1∣F∣νΓ˜∫Ω˜(t)V˜(z,t)dz.{V}_{\widetilde{\Gamma }}\left(z,t)=\widetilde{V}\cdot {\nu }_{\widetilde{\Gamma }}-\frac{1}{| {\mathscr{F}}| }{\nu }_{\widetilde{\Gamma }}\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}\left(z,t){\rm{d}}z.We will see in Section 5 that this modification will be convenient.Next, we transform (3.2) to a system on a domain F{\mathscr{F}}surrounded by a fixed surface G{\mathscr{G}}via the Hanzawa transform, where the strategy is due to Köhne et al. [12, Sec. 2] (cf. Prüss and Siminett [22]). For the necessary geometric background, we refer to Chapter 2 in [22]. Recall that the second-order bundle of Γ˜\widetilde{\Gamma }is given by N2Γ˜≔{(p,νΓ˜(p),∇Γ˜νΓ˜(p)):p∈Γ˜},{{\mathcal{N}}}^{2}\widetilde{\Gamma }:= \left\{\left(p,{\nu }_{\widetilde{\Gamma }}\left(p),{\nabla }_{\widetilde{\Gamma }}{\nu }_{\widetilde{\Gamma }}\left(p))\hspace{0.33em}:\hspace{0.33em}p\in \widetilde{\Gamma }\right\},where νΓ˜{\nu }_{\widetilde{\Gamma }}is the surface gradient on Γ˜\widetilde{\Gamma }. In addition, let us write dH{d}_{H}to denote the Hausdorff distance between the closed subsets K1,K2⊂R3{K}_{1},{K}_{2}\subset {{\mathbb{R}}}^{3}, which is defined by dH(K1,K2)≔max(supa∈K1dist(a,K2),supb∈K2dist(b,K1)).{d}_{H}\left({K}_{1},{K}_{2}):= \max \left(\mathop{\sup }\limits_{a\in {K}_{1}}{\rm{dist}}\hspace{0.33em}\left(a,{K}_{2}),\mathop{\sup }\limits_{b\in {K}_{2}}{\rm{dist}}\hspace{0.33em}\left(b,{K}_{1})\right).Then, the unknown surface Γ˜\widetilde{\Gamma }can be approximated by a real analytic hypersurface G{\mathscr{G}}in the sense that dH(N2Γ˜,N2G){d}_{H}\left({{\mathcal{N}}}^{2}\widetilde{\Gamma },{{\mathcal{N}}}^{2}{\mathscr{G}})is as small as we wish, i.e., for each η>0\eta \gt 0, we can find a real analytic closed hypersurface G{\mathscr{G}}such that dH(N2Γ˜,N2G)≤η{d}_{H}\left({{\mathcal{N}}}^{2}\widetilde{\Gamma },{{\mathcal{N}}}^{2}{\mathscr{G}})\le \eta . Since it is well known that the hypersurface G{\mathscr{G}}admits a tubular neighborhood, there exists some positive constant d0{{\mathsf{d}}}_{0}such that the mapping Λ:G×(−d0,d0)→R3\Lambda :{\mathscr{G}}\times \left(-{{\mathsf{d}}}_{0},{{\mathsf{d}}}_{0})\to {{\mathbb{R}}}^{3}defined by Λ(p,r)≔p+rνG(p),p∈G,∣r∣<d0\Lambda \left(p,r):= p+r{\nu }_{{\mathscr{G}}}\left(p),\hspace{1.0em}p\in {\mathscr{G}},| r| \lt {{\mathsf{d}}}_{0}is a diffeomorphism from G×(−d0,d0){\mathscr{G}}\times \left(-{{\mathsf{d}}}_{0},{{\mathsf{d}}}_{0})onto R(Λ){\mathsf{R}}\left(\Lambda ). Here, R(Λ){\mathsf{R}}\left(\Lambda )denotes the range of Λ\Lambda . Then, the inverse Λ−1:R(Λ)→G×(−d0,d0){\Lambda }^{-1}:{\mathsf{R}}\left(\Lambda )\to {\mathscr{G}}\times \left(-{{\mathsf{d}}}_{0},{{\mathsf{d}}}_{0})is conveniently decomposed as Λ−1(y)=(ΠG(y),dG(y)){\Lambda }^{-1}(y)=\left({\Pi }_{{\mathscr{G}}}(y),{d}_{{\mathscr{G}}}(y))for y∈R(Λ)y\in {\mathsf{R}}\left(\Lambda ), where ΠG(y){\Pi }_{{\mathscr{G}}}(y)and dG(y){d}_{{\mathscr{G}}}(y)stand for the orthogonal projection of yyonto G{\mathscr{G}}and the signed distance from yyonto G{\mathscr{G}}; so ∣dG(y)∣=dist(y,G)| {d}_{{\mathscr{G}}}(y)| ={\rm{dist}}\hspace{0.33em}(y,{\mathscr{G}})and dG(y)<0{d}_{{\mathscr{G}}}(y)\lt 0if and only if y∈Fy\in {\mathscr{F}}. Noting the compactness of G{\mathscr{G}}, there exists a radius rG>0{r}_{{\mathscr{G}}}\gt 0such that for each point p∈Gp\in {\mathscr{G}}there are balls B(y,rG)⊂FB(y,{r}_{{\mathscr{G}}})\subset {\mathscr{F}}satisfying G∩B(y,rG)¯={p}{\mathscr{G}}\cap \overline{B(y,{r}_{{\mathscr{G}}})}=\left\{p\right\}. Choosing rG{r}_{{\mathscr{G}}}maximal, it holds rG>d0{r}_{{\mathscr{G}}}\gt {{\mathsf{d}}}_{0}. In the following, we fix d0=rG/2{{\mathsf{d}}}_{0}={r}_{{\mathscr{G}}}\hspace{0.1em}\text{/}\hspace{0.1em}2and d=d0/3{\mathsf{d}}={{\mathsf{d}}}_{0}\hspace{0.1em}\text{/}\hspace{0.1em}3.We write the derivatives of dG(y){d}_{{\mathscr{G}}}(y)and ΠG(y){\Pi }_{{\mathscr{G}}}(y)by ∇dG(y)=νG(ΠG(y)),DΠG(y)=M0(dG(y))PG(ΠG(y)),\nabla {d}_{{\mathscr{G}}}(y)={\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)),\hspace{1.0em}D{\Pi }_{{\mathscr{G}}}(y)={M}_{0}\left({d}_{{\mathscr{G}}}(y)){{\mathcal{P}}}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)),respectively. Here, M0(r)≔(I−rLG)−1{M}_{0}\left(r):= {\left(I-r{L}_{{\mathscr{G}}})}^{-1}is the Weingarten tensor LG≔−∇GνG{L}_{{\mathscr{G}}}:= -{\nabla }_{{\mathscr{G}}}{\nu }_{{\mathscr{G}}}and PG(p)=I−νG(p)⊗νG(p){{\mathcal{P}}}_{{\mathscr{G}}}\left(p)=I-{\nu }_{{\mathscr{G}}}\left(p)\otimes {\nu }_{{\mathscr{G}}}\left(p)represents the orthogonal projection onto the tangent space of G{\mathscr{G}}at p∈Gp\in {\mathscr{G}}. Here, it holds ∣M0(r)∣≤(1−r∣LG∣)≤3| {M}_{0}\left(r)| \le \left(1-r| {L}_{{\mathscr{G}}}| )\le 3for all ∣r∣≤2rG/3| r| \le 2{r}_{{\mathscr{G}}}\hspace{0.1em}\text{/}\hspace{0.1em}3. Using the mapping Λ\Lambda , we may approximate the unknown free surface Γ˜(t)\widetilde{\Gamma }\left(t)over G{\mathscr{G}}by means of a height function hhvia Γ˜(t)≔{p+h(p,t)νG(p):p∈G}\widetilde{\Gamma }\left(t):= \left\{p+h\left(p,t){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {\mathscr{G}}\right\}for small t≥0t\ge 0, at least. We extend this diffeomorphism to all of F¯\overline{{\mathscr{F}}}and define the Hanzawa transform by Ξh(y,t)≔y+χdG(y)dh(ΠG(y),t)νG(ΠG(y))≕y+ξh(y,t),{\Xi }_{h}(y,t):= y+\chi \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t){\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\hspace{0.33em}=: \hspace{0.33em}y+{\xi }_{h}(y,t),where χ∈C∞(R)\chi \in {C}^{\infty }\left({\mathbb{R}})is a cutoff function such that 0≤χ≤10\le \chi \le 1, 1<∣χ′∣<31\lt | \chi ^{\prime} | \lt 3, χ(r)=1\chi \left(r)=1for ∣r∣≤1| r| \le 1, and χ(r)=0\chi \left(r)=0for ∣r∣>2| r| \gt 2. According to the definition of Ξh{\Xi }_{h}, we obtain Ξh(y,t)=zif∣dG(y)∣>2d,ΠG(Ξh(y,t))=ΠG(y)if∣dG(y)∣<d,dG(Ξh(y,t))=dG(y)+χdGdh(ΠG(y),t)if∣dG(y)∣<2d,\begin{array}{rcll}{\Xi }_{h}(y,t)& =& z& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \gt 2{\mathsf{d}},\\ {\Pi }_{{\mathscr{G}}}\left({\Xi }_{h}(y,t))& =& {\Pi }_{{\mathscr{G}}}(y)& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \lt {\mathsf{d}},\\ {d}_{{\mathscr{G}}}\left({\Xi }_{h}(y,t))& =& {d}_{{\mathscr{G}}}(y)+\chi \left(\frac{{d}_{{\mathscr{G}}}}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t)& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \lt 2{\mathsf{d}},\end{array}as well as Ξh−1(y,t)=y−h(ΠG(y),t)νG(ΠG(y))if∣dG(y)∣<d.{\Xi }_{h}^{-1}(y,t)=y-h\left({\Pi }_{{\mathscr{G}}}(y),t){\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\hspace{1.0em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \lt {\mathsf{d}}.By using the transform Ξh{\Xi }_{h}, we define the pull backs of V˜\widetilde{V}and P˜\widetilde{P}by u(y,t)≔V˜(Ξh(y,t),t),q(y,t)≔P˜(Ξh(y,t),t),y∈F,t>0,u(y,t):= \widetilde{V}\left({\Xi }_{h}(y,t),t),\hspace{1.0em}q(y,t):= \widetilde{P}\left({\Xi }_{h}(y,t),t),\hspace{1.0em}y\in {\mathscr{F}},\hspace{1.0em}t\gt 0,respectively. Then, we observe that (u,q,h)\left(u,q,h)satisfies the following system: (3.3)∂tu−μℒ(h)u+2ω(e3×u)+G(h)q=R(u,h),inF,G(h)⋅u=0,inF,(μ(G(h)u+[G(h)u]⊤)−qI)νΓ˜(h)=σHΓ˜(h)+ω22∣z′∣2+p0νΓ˜(h),onG,∂th−u⋅νG=−u⋅a(h)−1∣F∣(νG−a(h))⋅∫Ω˜(t)V˜dy,onG,u(0)=u0,inF,h(0)=h0,onG.\left\{\begin{array}{ll}{\partial }_{t}u-\mu {\mathcal{ {\mathcal L} }}\left(h)u+2\omega \left({e}_{3}\times u)+{\mathcal{G}}\left(h)q=R\left(u,h),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathscr{F}},\\ {\mathcal{G}}\left(h)\cdot u=0,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathscr{F}},\\ (\mu \left({\mathcal{G}}\left(h)u+{\left[{\mathcal{G}}\left(h)u]}^{\top })-qI){\nu }_{\widetilde{\Gamma }}\left(h)=\left(\sigma {{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)+\frac{{\omega }^{2}}{2}| z^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{\widetilde{\Gamma }}\left(h),& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}},\\ {\partial }_{t}h-u\cdot {\nu }_{{\mathscr{G}}}=-u\cdot a\left(h)-\frac{1}{| {\mathscr{F}}| }\left({\nu }_{{\mathscr{G}}}-a\left(h))\cdot \mathop{\displaystyle \int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}{\rm{d}}y,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}},\\ u\left(0)={u}_{0},& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathscr{F}},\\ h\left(0)={h}_{0},& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}}.\end{array}\right.Here, ℒ(h){\mathcal{ {\mathcal L} }}\left(h), G(h){\mathcal{G}}\left(h), and HΓ˜(h){{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)are the transformed Laplacian, gradient, and doubled mean curvature, respectively, while the functions R(u,h)R\left(u,h)and a(h)a\left(h)are defined below. Furthermore, it holds DΞh=I+Dξh,(DΞh)−1=I−(I+Dξh)−1ξh≕I−[M1(h)]⊤,D{\Xi }_{h}=I+D{\xi }_{h},\hspace{1.0em}{\left(D{\Xi }_{h})}^{-1}=I-{\left(I+D{\xi }_{h})}^{-1}{\xi }_{h}\hspace{0.33em}=: \hspace{0.33em}I-{\left[{M}_{1}\left(h)]}^{\top },where Dξh(y,t)=1aχ′dG(y)dh(ΠG(y),t)νG(ΠG(y))⊗νG(ΠG(y))+χdG(y)dνG(y)⊗M0(ΠG(y))∇Gh(ΠG(y),t)−χdG(y)dh(ΠG(y),t)LG(ΠG(y))M0(ΠG(y))PG(ΠG(y))\begin{array}{rcl}D{\xi }_{h}(y,t)& =& \frac{1}{a}\chi ^{\prime} \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t){\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\displaystyle \otimes {\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\\ & & +\chi \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right){\nu }_{{\mathscr{G}}}(y)\displaystyle \otimes {M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){\nabla }_{{\mathscr{G}}}h\left({\Pi }_{{\mathscr{G}}}(y),t)\\ & & -\chi \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t){L}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)){M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){{\mathcal{P}}}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\end{array}if ∣dG(y)∣<2d| {d}_{{\mathscr{G}}}(y)| \lt 2{\mathsf{d}}, while Dξh(y,t)=0D{\xi }_{h}(y,t)=0if ∣dG(y)∣>2d| {d}_{{\mathscr{G}}}(y)| \gt 2{\mathsf{d}}. In particular, if ∣dG(y)∣<d| {d}_{{\mathscr{G}}}(y)| \lt {\mathsf{d}}, we have Dξh(y,t)=νG(y)⊗M0(ΠG(y))∇Gh(ΠG(y),t)−h(ΠG(y),t)LG(ΠG(y))M0(ΠG(y))PG(ΠG(y)).D{\xi }_{h}(y,t)={\nu }_{{\mathscr{G}}}(y)\otimes {M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){\nabla }_{{\mathscr{G}}}h\left({\Pi }_{{\mathscr{G}}}(y),t)-h\left({\Pi }_{{\mathscr{G}}}(y),t){L}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)){M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){{\mathcal{P}}}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)).On the basis of these representations, we find that (I+Dξh)\left(I+D{\xi }_{h})is boundedly invertible if ∣h∣L∞(G)<13mind,1∣LG∣L∞(G)and∣∇Gh∣L∞(G)<13.| h{| }_{{L}^{\infty }\left({\mathscr{G}})}\lt \frac{1}{3}\min \left({\mathsf{d}},\frac{1}{| {L}_{{\mathscr{G}}}{| }_{{L}^{\infty }\left({\mathscr{G}})}}\right)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}| {\nabla }_{{\mathscr{G}}}h{| }_{{L}^{\infty }\left({\mathscr{G}})}\lt \frac{1}{3}.Then we can write [∇P˜]∘Ξh=G(h)q=[DΞh−1∘Ξh]⊤∇q=[(DΞh)−1]⊤∇q=∇q−M1(h)∇q,[divV˜]∘Ξh=G(h)⋅u=((I−M1(h))∇⋅)u,\begin{array}{rcl}\left[\nabla \widetilde{P}]\circ {\Xi }_{h}& =& {\mathcal{G}}\left(h)q={\left[D{\Xi }_{h}^{-1}\circ {\Xi }_{h}]}^{\top }\nabla q={\left[{\left(D{\Xi }_{h})}^{-1}]}^{\top }\nabla q=\nabla q-{M}_{1}\left(h)\nabla q,\\ {[}{\rm{div}}\hspace{0.33em}\widetilde{V}]\circ {\Xi }_{h}& =& {\mathcal{G}}\left(h)\cdot u=\left(\left(I-{M}_{1}\left(h))\nabla \cdot )u,\end{array}and [ΔV˜]∘Ξh=ℒ(h)u=[(I−M1(h))∇]⋅[(I−M1(h))∇h]=Δu−[M1(h)+[M1(h)]⊤−M1(h)[M1(h)]⊤]:∇2u+[[(I−M1(h)):∇M1(h)]⋅∇]u≕Δu−M2(h):∇2u−M3(h)⋅∇u.\begin{array}{rcl}\left[\Delta \widetilde{V}]\circ {\Xi }_{h}& =& {\mathcal{ {\mathcal L} }}\left(h)u\\ & =& \left[\left(I-{M}_{1}\left(h))\nabla ]\cdot \left[\left(I-{M}_{1}\left(h))\nabla h]\\ & =& \Delta u-\left[{M}_{1}\left(h)+{\left[{M}_{1}\left(h)]}^{\top }-{M}_{1}\left(h){\left[{M}_{1}\left(h)]}^{\top }]:{\nabla }^{2}u+\left[\left[\left(I-{M}_{1}\left(h)):\nabla {M}_{1}\left(h)]\cdot \nabla ]u\\ & =: & \Delta u-{M}_{2}\left(h):{\nabla }^{2}u-{M}_{3}\left(h)\cdot \nabla u.\end{array}Here, we have used the notation A:B=∑i,j=13aijbij=tr(AB⊤)A:B=\mathop{\sum }\limits_{i,j=1}^{3}{a}_{ij}{b}_{ij}={\rm{tr}}\left(A{B}^{\top }). In our analysis, it is required to obtain another representation formula for [divV˜]∘Ξh\left[{\rm{div}}\hspace{0.33em}\widetilde{V}]\circ {\Xi }_{h}. To this end, we use the L2{L}^{2}inner product in Ω˜(t)\widetilde{\Omega }\left(t), which is denoted by (⋅,⋅)Ω˜(t){\left(\cdot ,\cdot )}_{\widetilde{\Omega }\left(t)}. For any test function φ˜∈Cc∞(Ω˜(t))\widetilde{\varphi }\in {C}_{c}^{\infty }\left(\widetilde{\Omega }\left(t)), we write φ(y)=φ˜(z)\varphi (y)=\widetilde{\varphi }\left(z). Here, Cc∞(D){C}_{c}^{\infty }\left(D)is the set of all C∞{C}^{\infty }-functions on R3{{\mathbb{R}}}^{3}, which have compact supports contained in D⊂R3D\subset {{\mathbb{R}}}^{3}. In addition, the Jacobian of the transform Ξh(y,t){\Xi }_{h}(y,t)is denoted by J=J(h){\mathsf{J}}={\mathsf{J}}\left(h). Recalling the expression of DξhD{\xi }_{h}, we shall write J(h)=1+J0(h){\mathsf{J}}\left(h)=1+{{\mathsf{J}}}_{0}\left(h)with some function J0(h){{\mathsf{J}}}_{0}\left(h)that vanishes at h=0h=0. Then, we see that (divV˜,φ˜)Ω˜(t)=−(V˜,∇φ˜)Ω˜(t)=−(Ju,(I−M1(h))∇φ)F=(div((I−[M1(h)]⊤)Ju),φ)F=(J−1div((I−[M1(h)]⊤)Ju),φ˜)Ω˜(t),\begin{array}{rcl}{\left({\rm{div}}\widetilde{V},\widetilde{\varphi })}_{\widetilde{\Omega }\left(t)}& =& -{\left(\widetilde{V},\nabla \widetilde{\varphi })}_{\widetilde{\Omega }\left(t)}\\ & =& -{\left({\mathsf{J}}u,\left(I-{M}_{1}\left(h))\nabla \varphi )}_{{\mathscr{F}}}\\ & =& {\left({\rm{div}}\left(\left(I-{\left[{M}_{1}\left(h)]}^{\top }){\mathsf{J}}u),\varphi )}_{{\mathscr{F}}}\\ & =& {\left({{\mathsf{J}}}^{-1}{\rm{div}}\left(\left(I-{\left[{M}_{1}\left(h)]}^{\top }){\mathsf{J}}u),\widetilde{\varphi })}_{\widetilde{\Omega }\left(t)},\end{array}where (⋅,⋅)F{\left(\cdot ,\cdot )}_{{\mathscr{F}}}is the L2{L}^{2}inner product in F{\mathscr{F}}. This gives [divV˜]∘Ξh=divu−M1(h):∇u=J−1(div((I−[M1(h)]⊤)Ju)),\left[{\rm{div}}\hspace{0.33em}\widetilde{V}]\circ {\Xi }_{h}={\rm{div}}\hspace{0.33em}u-{M}_{1}\left(h):\nabla u={{\mathsf{J}}}^{-1}({\rm{div}}\hspace{0.33em}\left(\left(I-{\left[{M}_{1}\left(h)]}^{\top }){\mathsf{J}}u)),i.e., the divergence free condition (3.2)2{\left(3.2)}_{2}is rewritten as follows: divu=−J0(h)divu+(1+J0(h))M1(h):∇u=div((1+J0(h))[M1(h)]⊤u).{\rm{div}}\hspace{0.33em}u=-{{\mathsf{J}}}_{0}\left(h){\rm{div}}\hspace{0.33em}u+\left(1+{{\mathsf{J}}}_{0}\left(h)){M}_{1}\left(h):\nabla u={\rm{div}}\hspace{0.33em}(\left(1+{{\mathsf{J}}}_{0}\left(h)){\left[{M}_{1}\left(h)]}^{\top }u).Next, we note that [∂tV˜]∘Ξh=∂tu−[(∇V˜)∘Ξh](∂tΞh)=∂tu−[[DΞh−1∘Ξh]⊤∇u](∂tΞh)=∂tu−∇u[(I+Dξh)−1∂tξh]≕∂tu−M4(h)∇u,\begin{array}{rcl}\left[{\partial }_{t}\widetilde{V}]\circ {\Xi }_{h}& =& {\partial }_{t}u-\left[\left(\nabla \widetilde{V})\circ {\Xi }_{h}]\left({\partial }_{t}{\Xi }_{h})\\ & =& {\partial }_{t}u-\left[{\left[D{\Xi }_{h}^{-1}\circ {\Xi }_{h}]}^{\top }\nabla u]\left({\partial }_{t}{\Xi }_{h})\\ & =& {\partial }_{t}u-\nabla u\left[{\left(I+D{\xi }_{h})}^{-1}{\partial }_{t}{\xi }_{h}]\\ & =: & {\partial }_{t}u-{M}_{4}\left(h)\nabla u,\end{array}which implies R(u,h)=−(u⋅G(h)u)+M4(h)∇u.R\left(u,h)=-\left(u\cdot {\mathcal{G}}\left(h)u)+{M}_{4}\left(h)\nabla u.Furthermore, it holds νΓ˜=b(h)(νG−a(h)),a(h)=M0(h)∇Gh,b(h)=11+∣a(h)∣2,M0(h)=(I−hLG)−1,{\nu }_{\widetilde{\Gamma }}=b\left(h)\left({\nu }_{{\mathscr{G}}}-a\left(h)),\hspace{1.0em}a\left(h)={M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h,\hspace{1.0em}b\left(h)=\frac{1}{\sqrt{1+| a\left(h){| }^{2}}},\hspace{1.0em}{M}_{0}\left(h)={\left(I-h{L}_{{\mathscr{G}}})}^{-1},and VΓ˜=(∂tΞh)⋅νΓ˜=∂th(νΓ˜⋅νG)=b(h)∂th.{V}_{\widetilde{\Gamma }}=\left({\partial }_{t}{\Xi }_{h})\cdot {\nu }_{\widetilde{\Gamma }}={\partial }_{t}h\left({\nu }_{\widetilde{\Gamma }}\cdot {\nu }_{{\mathscr{G}}})=b\left(h){\partial }_{t}h.Here, νG{\nu }_{{\mathscr{G}}}and a(h)a\left(h)are linearly independent. The term ∫Ω^(t)V˜(y,t)dy{\int }_{\widehat{\Omega }\left(t)}\widetilde{V}(y,t){\rm{d}}ycan be read as follows: ∫Ω˜(t)V˜dy=∫FuJ(h)dz=∫Fudz+∫FuJ0(h)dz.\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}{\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}u{\mathsf{J}}\left(h){\rm{d}}z=\mathop{\int }\limits_{{\mathscr{F}}}u{\rm{d}}z+\mathop{\int }\limits_{{\mathscr{F}}}u{{\mathsf{J}}}_{0}\left(h){\rm{d}}z.The doubled mean curvature HΓ˜{{\mathscr{H}}}_{\widetilde{\Gamma }}is given by (3.4)HΓ˜(h)=b(h)(tr[M0(h)(LG+∇Ga(h))]−b2(h)(M0(h)a(h))⋅([∇Ga(h)]a(h))).{{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)=b\left(h)({\rm{tr}}\left[{M}_{0}\left(h)\left({L}_{{\mathscr{G}}}+{\nabla }_{{\mathscr{G}}}a\left(h))]-{b}^{2}\left(h)\left({M}_{0}\left(h)a\left(h))\cdot \left(\left[{\nabla }_{{\mathscr{G}}}a\left(h)]a\left(h))).Its linearization at h=0h=0is given by (3.5)HΓ˜′(0)=trLG2+ΔG=HG2−2KG+ΔG,{{\mathscr{H}}}_{\widetilde{\Gamma }}^{^{\prime} }\left(0)={\rm{tr}}{L}_{{\mathscr{G}}}^{2}+{\Delta }_{{\mathscr{G}}}={{\mathscr{H}}}_{{\mathscr{G}}}^{2}-2{{\mathscr{K}}}_{{\mathscr{G}}}+{\Delta }_{{\mathscr{G}}},where ΔG{\Delta }_{{\mathscr{G}}}is the Laplace-Beltrami operator on G{\mathscr{G}}. Here and in the following, for sufficiently smooth functions a,b∈C(G){\mathsf{a}},{\mathsf{b}}\in C\left({\mathscr{G}})and F(a):G→Rk{\mathsf{F}}\left({\mathsf{a}}):{\mathscr{G}}\to {{\mathbb{R}}}^{k}, we use the notation F′(a)b≔ddsF(a+sb)∣s=0,{\mathsf{F}}^{\prime} \left({\mathsf{a}}){\mathsf{b}}:= \frac{{\rm{d}}}{{\rm{d}}s}{\mathsf{F}}\left({\mathsf{a}}+s{\mathsf{b}}){| }_{s=0},which denotes the first variation of F(a){\mathsf{F}}\left({\mathsf{a}}). We refer to [22, Ch. 2] for the derivations of (3.4) and (3.5).We now decompose the stress boundary condition into tangential and normal parts. Multiplying (3.3)3{\left(3.3)}_{3}with νG/b{\nu }_{{\mathscr{G}}}\hspace{0.1em}\text{/}\hspace{0.1em}b, we obtain q+σHΓ˜(h)+ω22∣z′∣2+p0=(μ(G(h)u+[G(h)u]⊤)(νG−a(h)))⋅νGq+\sigma {{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)+\frac{{\omega }^{2}}{2}| z^{\prime} {| }^{2}+{p}_{0}=(\mu \left({\mathcal{G}}\left(h)u+{\left[{\mathcal{G}}\left(h)u]}^{\top })\left({\nu }_{{\mathscr{G}}}-a\left(h)))\cdot {\nu }_{{\mathscr{G}}}for the normal part of (3.3)3{\left(3.3)}_{3}, while (3.6)PG(μ(G(h)u+[G(h)u]⊤)(νG−a(h)))=0{{\mathcal{P}}}_{{\mathscr{G}}}(\mu \left({\mathcal{G}}\left(h)u+{\left[{\mathcal{G}}\left(h)u]}^{\top })\left({\nu }_{{\mathscr{G}}}-a\left(h)))=0for the tangential part of (3.3)3{\left(3.3)}_{3}. It should be emphasized that (3.6) neither contains the pressure nor the curvature. Finally, we have ∣z′∣2−∣y′∣2−∂∂νG∣y′∣2h=h2((νG(1))2+(νG(2))2).| z^{\prime} {| }^{2}-| y^{\prime} {| }^{2}-\left(\frac{\partial }{\partial {\nu }_{{\mathscr{G}}}}| y^{\prime} {| }^{2}\right)h={h}^{2}({\left({\nu }_{{\mathscr{G}}}^{\left(1)})}^{2}+{\left({\nu }_{{\mathscr{G}}}^{\left(2)})}^{2}).Consequently, from the aforementioned discussion, problem (3.2) can be rewritten as follows: (3.7)∂tu−μΔu+2ω(e3×u)+∇q=Fu(u,q,h),in F,divu=Gd(u,h)=divGdiv(u,h),in F,PG(μ(∇u+[∇u]⊤)νG)=Guτ(u,h),on G,μ(∇u+[∇u]⊤)νG⋅νG−q+ℬGh=Guv(u,h)+G0(h),on G,∂th−u⋅νG+1∣F∣νG⋅∫Fudz=Fh(u,h)+F(u,h),on G,u(0)=u0,in F,h(0)=h0,on G.\left\{\begin{array}{ll}{\partial }_{t}u-\mu \Delta u+2\omega \left({e}_{3}\times u)+\nabla q={F}_{u}\left(u,q,h),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}u={G}_{d}\left(u,h)={\rm{div}}\hspace{0.33em}{G}_{{\rm{div}}}\left(u,h),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}(\mu \left(\nabla u+{\left[\nabla u]}^{\top }){\nu }_{{\mathscr{G}}})={G}_{u\tau }\left(u,h),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \mu \left(\nabla u+{\left[\nabla u]}^{\top }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-q+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h={G}_{uv}\left(u,h)+{G}_{0}\left(h),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}h-u\cdot {\nu }_{{\mathscr{G}}}+\frac{1}{| {\mathscr{F}}| }{\nu }_{{\mathscr{G}}}\cdot \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}u{\rm{d}}z={F}_{h}\left(u,h)+F\left(u,h),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ u\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ h\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Here, the right-hand members of (3.7) can be written as follows: Fu(u,q,h)=M4(h)∇u−u⋅(I−M1(h))∇u−μ(M2(h):∇2u+M3(h)⋅∇u)+M1(h)∇q,Gd(u,h)=−J0(h)divu+(1+J0(h))M1(h):∇u,Gdiv(u,h)=(1+J0(h))⊤M1(h)u,Guτ(u,h)=PG(μ(M1(h)∇u+[M1(h)∇u]⊤)(νG−M0(h)∇Gh))+PG((∇u+[∇u]⊤)M0(h)∇Gh),Guv(u,h)=−(∇u+[∇u]⊤)M0(h)∇Gh⋅νG+(M1(h)∇u+[M1(h)∇u]⊤)(νG−M0(h)∇Gh)⋅νGG0(h)=σ(HΓ˜(h)−HΓ˜′(0)h)+ω22{h2((νG(1))2+(νG(2))2)},Fh(u,h)=−(M0(h)∇Gh)⋅u,F(u,h)=1∣F∣a(h)⋅∫Fu(1+J0)dz−1∣F∣νG⋅∫FuJ0(h)dz.\begin{array}{rcl}{F}_{u}\left(u,q,h)& =& {M}_{4}\left(h)\nabla u-u\cdot \left(I-{M}_{1}\left(h))\nabla u-\mu ({M}_{2}\left(h):{\nabla }^{2}u+{M}_{3}\left(h)\cdot \nabla u)+{M}_{1}\left(h)\nabla q,\\ {G}_{d}\left(u,h)& =& -{{\mathsf{J}}}_{0}\left(h){\rm{div}}\hspace{0.33em}u+\left(1+{{\mathsf{J}}}_{0}\left(h)){M}_{1}\left(h):\nabla u,\\ {G}_{{\rm{div}}}\left(u,h)& =& {\left(1+{{\mathsf{J}}}_{0}\left(h))}^{\top }{M}_{1}\left(h)u,\\ {G}_{u\tau }\left(u,h)& =& {{\mathcal{P}}}_{{\mathscr{G}}}(\mu \left({M}_{1}\left(h)\nabla u+{\left[{M}_{1}\left(h)\nabla u]}^{\top })\left({\nu }_{{\mathscr{G}}}-{M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h))+{{\mathcal{P}}}_{{\mathscr{G}}}(\left(\nabla u+{\left[\nabla u]}^{\top }){M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h),\\ {G}_{uv}\left(u,h)& =& -\left(\nabla u+{\left[\nabla u]}^{\top }){M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h\cdot {\nu }_{{\mathscr{G}}}+\left({M}_{1}\left(h)\nabla u+{\left[{M}_{1}\left(h)\nabla u]}^{\top })\left({\nu }_{{\mathscr{G}}}-{M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h)\cdot {\nu }_{{\mathscr{G}}}\\ {G}_{0}\left(h)& =& \sigma \left({{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)-{{\mathscr{H}}}_{\widetilde{\Gamma }}^{^{\prime} }\left(0)h)+\frac{{\omega }^{2}}{2}\{{h}^{2}({\left({\nu }_{{\mathscr{G}}}^{\left(1)})}^{2}+{\left({\nu }_{{\mathscr{G}}}^{\left(2)})}^{2})\},\\ {F}_{h}\left(u,h)& =& -\left({M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h)\cdot u,\\ F\left(u,h)& =& \frac{1}{| {\mathscr{F}}| }a\left(h)\cdot \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}u\left(1+{{\mathsf{J}}}_{0}){\rm{d}}z-\frac{1}{| {\mathscr{F}}| }{\nu }_{{\mathscr{G}}}\cdot \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}u{{\mathsf{J}}}_{0}\left(h){\rm{d}}z.\end{array}Notice that the right-hand members of (3.7) vanish at (u,q,h)=(0,0,0)\left(u,q,h)=\left(0,0,0).4Maximal regularityTo simplify the notation, in the following, we write D(u)≔2−1(∇u+[∇u]⊤)D\left(u):= {2}^{-1}\left(\nabla u+{\left[\nabla u]}^{\top })for every vector field uu. The principal part of the linearized problem reads as follows: (4.1)∂tu−μΔu+2ω(e3×u)+∇q=fu,in F,divu=gd,in F,PG(2μD(u)νG)=guτ,on G,2μD(u)νG⋅νG−q+ℬGh=guv,on G,∂th−(P0Gu)⋅νG=fh,on G,u(0)=u0,in F,h(0)=h0,on G,\left\{\begin{array}{ll}{\partial }_{t}u-\mu \Delta u+2\omega \left({e}_{3}\times u)+\nabla q={f}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}u={g}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(u){\nu }_{{\mathscr{G}}})={g}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(u){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-q+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h={g}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}h-\left({P}_{0}^{{\mathscr{G}}}u)\cdot {\nu }_{{\mathscr{G}}}={f}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ u\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ h\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where we have set P0Gu=u−1∣F∣∫Fu(y,t)dy.{P}_{0}^{{\mathscr{G}}}u=u-\frac{1}{| {\mathscr{F}}| }\mathop{\int }\limits_{{\mathscr{F}}}u(y,t){\rm{d}}y.We define the function spaces Eδ(J;F)≔E1,δ(J;F)×E2,δ(J;F)×E3,δ(J;G)×E4,δ(J;G),E1,δ(J;F)≔Hδ1,p(J;Lq(F)3)∩Lδp(J;H2,q(F)3),E2,δ(J;F)≔Lδp(J;H˙1,q(F)),E3,δ(J;G)≔Fp,q,δ1/2−1/(2q)(J;Lq(G))∩Lδp(J;Bq,q1−1/q(G)),E4,δ(J;G)≔Fp,q,δ2−1/q(J;Lq(G))∩Hδ1,p(J;Bq,q2−1/q(G))∩Lδp(J;Bq,q3−1/q(G)),Fδ(J;F)≔F1,δ(J;F)×F2,δ(J;F)×F3,δ(J;F)2×F4,δ(J;G),F1,δ(J;F)≔Lδp(J;Lq(F)3),F2,δ(J;F)≔Hδ1,p(J;H˙−1,q(F))∩Lδp(J;DIq(F)),F3,δ(J;G)≔Fp,q,δ1/2−1/(2q)(J;Lq(G))∩Lδp(J;Bq,q1−1/q(G)),F4,δ(J;G)≔Fp,q,δ1−1/(2q)(J;Lq(G))∩Lδp(J;Bq,q2−1/q(G))\begin{array}{rcl}{{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {{\mathbb{E}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{E}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})\times {{\mathbb{E}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}}),\\ {{\mathbb{E}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {H}_{\delta }^{1,p}\left(J;\hspace{0.33em}{L}^{q}{\left({\mathscr{F}})}^{3})\cap {L}_{\delta }^{p}\left(J;{H}^{2,q}{\left({\mathscr{F}})}^{3}),\\ {{\mathbb{E}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {L}_{\delta }^{p}\left(J;\hspace{0.33em}{\dot{H}}^{1,q}\left({\mathscr{F}})),\\ {{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{1\hspace{0.1em}\text{/}2-1\text{/}\hspace{0.1em}\left(2q)}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{1-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})),\\ {{\mathbb{E}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {H}_{\delta }^{1,p}\left(J;\hspace{0.33em}{B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})),\\ {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {{\mathbb{F}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{F}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{F}}}_{3,\delta }{\left(J;{\mathscr{F}})}^{2}\times {{\mathbb{F}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}}),\\ {{\mathbb{F}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {L}_{\delta }^{p}\left(J;\hspace{0.33em}{L}^{q}{\left({\mathscr{F}})}^{3}),\\ {{\mathbb{F}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {H}_{\delta }^{1,p}\left(J;\hspace{0.33em}{\dot{H}}^{-1,q}\left({\mathscr{F}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{{\rm{DI}}}_{q}\left({\mathscr{F}})),\\ {{\mathbb{F}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{1\hspace{0.1em}\text{/}2-1\text{/}\hspace{0.1em}\left(2q)}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{1-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})),\\ {{\mathbb{F}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{1-1\hspace{0.1em}\text{/}\hspace{0.1em}\left(2q)}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\end{array}for J⊂R+J\subset {{\mathbb{R}}}_{+}. The main theorem of this section states that problem (4.1) has maximal regularity.Theorem 4.1Let 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. There exists a constant β0>0{\beta }_{0}\gt 0such that for all β≥β0\beta \ge {\beta }_{0}, problem (4.1) has a unique solution (u,q,TrG[q],h)∈eβtEδ(R+;F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {e}^{\beta t}{{\mathbb{E}}}_{\delta }\left({{\mathbb{R}}}_{+};\hspace{0.33em}{\mathscr{F}})if and only if(a)(u0,h0)∈Bq,p2(δ−1/p)(F)3×Bq,p2+δ−1/p−1/q(G)\left({u}_{0},{h}_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}});(b)(fu,gd,guτ,guv,fh)∈eβtFδ(R+;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {e}^{\beta t}{{\mathbb{F}}}_{\delta }\left({{\mathbb{R}}}_{+};\hspace{0.33em}{\mathscr{F}});(c)gd∣t=0=divu0{g}_{d}{| }_{t=0}={\rm{div}}\hspace{0.33em}{u}_{0};(d)guτ∣t=0=PG(2μD(u)νG){g}_{u\tau }{| }_{t=0}={{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(u){\nu }_{{\mathscr{G}}})if 1/p+1/(2q)<δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\lt \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2.Furthermore, the solution (u,q,TrG[q],h)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)enjoys the estimate∣e−βt(u,q,TrG[q],h)∣Eδ(R+;F)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣η0∣Bq,p2+δ−1/p−1/q(G)+∣e−βt(fu,gd,guτ,guv,fh)∣Fδ(R+;F))| {e}^{-\beta t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left({{\mathbb{R}}}_{+};{\mathscr{F}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {\eta }_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{-\beta t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left({{\mathbb{R}}}_{+};{\mathscr{F}})})with some constant C independent of ω\omega , β0{\beta }_{0}, β\beta , and TT.To prove Theorem 4.1, we consider the corresponding resolvent problem: (4.2)λu^−μΔu^+2ω(e3×u^)+∇q^=f^u,in F,divu^=g^d,in F,PG(2μD(u^)νG)=g^uτ,on G,2μD(u^)νG⋅νG−q^+ℬGh^=g^uv,on G,λh^−(P0Gu^)⋅νG=f^h,on G,\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q}={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}={\widehat{g}}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})={\widehat{g}}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h}={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.which can be obtained by the Laplace transform, with respect to time tt, applied to the system (4.1). Here, λ\lambda is the resolvent parameter varying in Σε,λ0≔{λ∈C:∣argλ∣≤π−ε,∣λ∣≥λ0}{\Sigma }_{\varepsilon ,{\lambda }_{0}}:= \left\{\lambda \in {\mathbb{C}}\hspace{0.33em}:\hspace{0.33em}| \arg \lambda | \le \pi -\varepsilon ,| \lambda | \ge {\lambda }_{0}\right\}for ε∈(0,π/2)\varepsilon \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2)and λ0>0{\lambda }_{0}\gt 0. By using the result obtained by Shibata [27, Thm. 4.8], we show the existence of ℛ{\mathcal{ {\mathcal R} }}-bounded solution operator families for (4.2). This section is mainly devoted to the proof of the following lemma.Lemma 4.2Assume 1<q<∞1\lt q\lt \infty and 0<ε<π/20\lt \varepsilon \lt \pi \hspace{0.1em}\text{/}\hspace{0.1em}2. LetXq≔(f^u,g^d,g^uτ,g^uv,f^h):f^u∈Lq(F)3,g^d∈DIq(F),g^uτ∈H1,q(F)2,g^uv∈H1,q(F),f^h∈Bq,q2−1/q(G),Xq≔F=(F1,…,F9):F1,F4∈Lq(F)3,F2∈Lq(F),F3∈H1,q(F),F5,F7∈Lq(F)2,F6,F8∈H1,q(F)2,F9∈Bq,q2−1/q(G)..\begin{array}{rcl}{X}_{q}& := & \left\{({\widehat{f}}_{u},{\widehat{g}}_{d},{\widehat{g}}_{u\tau },{\widehat{g}}_{uv},{\widehat{f}}_{h})\hspace{0.33em}:\hspace{0.33em}\begin{array}{c}{\widehat{f}}_{u}\in {L}^{q}{\left({\mathscr{F}})}^{3},{\widehat{g}}_{d}\in {{\rm{DI}}}_{q}\left({\mathscr{F}}),{\widehat{g}}_{u\tau }\in {H}^{1,q}{\left({\mathscr{F}})}^{2},\\ {\widehat{g}}_{uv}\in {H}^{1,q}\left({\mathscr{F}}),{\widehat{f}}_{h}\in {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})\end{array}\right\},\\ {{\mathcal{X}}}_{q}& := & \left\{F=\left({F}_{1},\ldots ,{F}_{9})\hspace{0.33em}:\hspace{0.33em}\begin{array}{c}{F}_{1},{F}_{4}\in {L}^{q}{\left({\mathscr{F}})}^{3},{F}_{2}\in {L}^{q}\left({\mathscr{F}}),{F}_{3}\in {H}^{1,q}\left({\mathscr{F}}),\\ {F}_{5},{F}_{7}\in {L}^{q}{\left({\mathscr{F}})}^{2},{F}_{6},{F}_{8}\in {H}^{1,q}{\left({\mathscr{F}})}^{2},{F}_{9}\in {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}).\end{array}\right\}.\end{array}Then there exists a constant λ1≥C0max(ω2,1){\lambda }_{1}\ge {C}_{0}\max \left({\omega }^{2},1)with some constant C0{C}_{0}and families of operators Uω(λ){{\mathcal{U}}}_{\omega }\left(\lambda ), Pω(λ){{\mathcal{P}}}_{\omega }\left(\lambda ), and ℋω(λ){{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )withUω(λ)∈Hol(Σε,λ1;ℒ(Xq,H2,q(F)3)),Qω(λ)∈Hol(Σε,λ1;ℒ(Xq,H1,q(F))),ℋω(λ)∈Hol(Σε,λ1;ℒ(Xq,Bq,q3−1/q(G))),\begin{array}{rcl}{{\mathcal{U}}}_{\omega }\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{1}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{2,q}{\left({\mathscr{F}})}^{3})),\\ {{\mathcal{Q}}}_{\omega }\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{1}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{1,q}\left({\mathscr{F}}))),\\ {{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{1}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))),\end{array}such that (u^,q^,η^)≔(Uω(λ),Pω(λ),ℋω(λ))FλG\left(\widehat{u},\widehat{q},\widehat{\eta }):= \left({{\mathcal{U}}}_{\omega }\left(\lambda ),{{\mathcal{P}}}_{\omega }\left(\lambda ),{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )){F}_{\lambda }Gis the unique solution to (4.2), where G and Fλ{F}_{\lambda }are given byG≔(f^u,g^d,G(g^d),g^uτ,g^uv,f^h),FλG≔(f^u,λ1/2g^d,g^d,λG(g^d),λ1/2g^uτ,g^uτ,λ1/2g^uv,g^uv,f^h).\begin{array}{rcl}G& := & ({\widehat{f}}_{u},{\widehat{g}}_{d},{\mathsf{G}}\left({\widehat{g}}_{d}),{\widehat{g}}_{u\tau },{\widehat{g}}_{uv},{\widehat{f}}_{h}),\\ {F}_{\lambda }G& := & ({\widehat{f}}_{u},{\lambda }^{1\text{/}2}{\widehat{g}}_{d},{\widehat{g}}_{d},\lambda {\mathsf{G}}\left({\widehat{g}}_{d}),{\lambda }^{1\text{/}2}{\widehat{g}}_{u\tau },{\widehat{g}}_{u\tau },{\lambda }^{1\text{/}2}{\widehat{g}}_{uv},{\widehat{g}}_{uv},{\widehat{f}}_{h}).\end{array}Besides, it holds(4.3)ℛXq→H2−j,q(F)3({(τ∂τ)ℓ(λj/2Uω(λ)):λ∈Σε,λ1})≤c,ℛXq→Lq(F)3({(τ∂τ)ℓ∇Qω(λ):λ∈Σε,λ1})≤c,ℛXq→Bq,q3−1/q−k(G)({(τ∂τ)ℓ(λkℋω(λ)):λ∈Σε,λ1})≤c\begin{array}{rcl}{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {H}^{2-j,q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{j\text{/}2}{{\mathcal{U}}}_{\omega }\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {L}^{q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\nabla {{\mathcal{Q}}}_{\omega }\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {B}_{q,q}^{3-1\text{/}q-k}\left({\mathscr{G}})}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{k}{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})& \le & c\end{array}for ℓ=0,1\ell =0,1, j=0,1,2j=0,1,2, k=0,1k=0,1, and τ≔Imλ\tau := {\rm{Im}}\hspace{0.33em}\lambda , where c is independent of ω\omega .ProofAccording to Shibata [27, Thm. 4.8], we know the existence of ℛ{\mathcal{ {\mathcal R} }}-bounded solution operator for the following problem: (4.4)λu^−μΔu^+∇q^=f^u,in F,divu^=g^d,in F,PG(2μD(u^)νG)=g^uτ,on G,2μD(u^)νG⋅νG−q^−σH′(0)h^=g^uv,on G,λh^−(P0Gu^)⋅νG=f^h,on G.\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+\nabla \widehat{q}={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}={\widehat{g}}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})={\widehat{g}}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}-\sigma {\mathscr{H}}^{\prime} \left(0)\widehat{h}={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.More precisely, he prove that there exist a constant λ0{\lambda }_{0}and families of operators U0(λ){{\mathcal{U}}}_{0}\left(\lambda ), P0(λ){{\mathcal{P}}}_{0}\left(\lambda ), and ℋ0(λ){{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )with U0(λ)∈Hol(Σε,λ0;ℒ(Xq,H2,q(F)3)),Q0(λ)∈Hol(Σε,λ0;ℒ(Xq,H1,q(F))),ℋ0(λ)∈Hol(Σε,λ0;ℒ(Xq,Bq,q3−1/q(G)))\begin{array}{rcl}{{\mathcal{U}}}_{0}\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{0}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{2,q}{\left({\mathscr{F}})}^{3})),\\ {{\mathcal{Q}}}_{0}\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{0}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{1,q}\left({\mathscr{F}}))),\\ {{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{0}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})))\end{array}such that for every λ∈Σε,λ0\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}and G∈XqG\in {X}_{q}, the triplet (u^,q^,h^)=(U0(λ),Q0(λ),ℋ0(λ))FλG\left(\widehat{u},\widehat{q},\widehat{h})=\left({{\mathcal{U}}}_{0}\left(\lambda ),{{\mathcal{Q}}}_{0}\left(\lambda ),{{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )){F}_{\lambda }Gis a unique solution to problem (4.4) satisfying (4.5)ℛXq→H2−j,q(F)3({(τ∂τ)ℓ(λj/2U0(λ)):λ∈Σε,λ0})≤c0,ℛXq→Lq(F)3({(τ∂τ)ℓ∇Q0(λ):λ∈Σε,λ0})≤c0,ℛXq→Bq,q3−1/q−k(G)({(τ∂τ)ℓ(λkℋ0(λ)):λ∈Σε,λ0})≤c0\begin{array}{rcl}{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {H}^{2-j,q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{j\text{/}2}{{\mathcal{U}}}_{0}\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}\})& \le & {c}_{0},\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {L}^{q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\nabla {{\mathcal{Q}}}_{0}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}\})& \le & {c}_{0},\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {B}_{q,q}^{3-1\text{/}q-k}\left({\mathscr{G}})}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{k}{{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}\})& \le & {c}_{0}\end{array}for ℓ=0,1\ell =0,1, j=0,1,2j=0,1,2, k=0,1k=0,1, and τ≔Imλ\tau := {\rm{Im}}\hspace{0.33em}\lambda . Here, λ0{\lambda }_{0}and c0{c}_{0}are independent of ω\omega . Then we see that the solution (u^,q^,h^)\left(\widehat{u},\widehat{q},\widehat{h})of (4.4) satisfies λu^−μΔu^+2ω(e3×u^)+∇q^=f^u+2ω(e3×u^),in F,divu^=g^d,in F,PG(2μD(u^)νG)=g^uτ,on G,2μD(u^)νG⋅νG−q^+ℬGh^=g^uv+ω2(y′⋅νG)h^,on G,λh^−(P0Gu^)⋅νG=f^h,on G.\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q}={\widehat{f}}_{u}+2\omega \left({e}_{3}\times \widehat{u}),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}={\widehat{g}}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})={\widehat{g}}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h}={\widehat{g}}_{uv}+{\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}})\widehat{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Now we define R1(λ)G≔−2ω(e3×U0(λ)FλG),R1(λ)F≔−2ω(e3×U0(λ)F),R2(λ)G≔−ω2(y′⋅νG)ℋ0(λ)FλG,R2(λ)F≔−ω2(y′⋅νG)ℋ0(λ)F\begin{array}{rcl}{R}_{1}\left(\lambda )G& := & -2\omega \left({e}_{3}\times {{\mathcal{U}}}_{0}\left(\lambda ){F}_{\lambda }G),\hspace{1.0em}{{\mathscr{R}}}_{1}\left(\lambda )F:= -2\omega \left({e}_{3}\times {{\mathcal{U}}}_{0}\left(\lambda )F),\\ {R}_{2}\left(\lambda )G& := & -{\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}}){{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda ){F}_{\lambda }G,\hspace{1.0em}{{\mathscr{R}}}_{2}\left(\lambda )F:= -{\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}}){{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )F\end{array}for G∈XqG\in {X}_{q}and F∈XqF\in {{\mathcal{X}}}_{q}. Setting R(λ)≔(R1(λ),0,0,R2(λ),0)R\left(\lambda ):= \left({R}_{1}\left(\lambda ),0,0,{R}_{2}\left(\lambda ),0)and R(λ)≔(R1(λ),0,0,R2(λ),0){\mathscr{R}}\left(\lambda ):= \left({{\mathscr{R}}}_{1}\left(\lambda ),0,0,{{\mathscr{R}}}_{2}\left(\lambda ),0), we have the relation (4.6)R(λ)=R(λ)Fλ,R\left(\lambda )={\mathscr{R}}\left(\lambda ){F}_{\lambda },which maps from Xq{X}_{q}to Xq{{\mathcal{X}}}_{q}. Since it holds ∣2ω(e3×u^)∣Lq(F)≤2ω∣u^∣Lq(F),∣ω2(y′⋅νG)h^∣H1,q(F)≤Cω2∣y′⋅νG∣H1,∞(F)∣h^∣H1,q(F)≤Cq,Gω2∣h^∣Bq,q2−1/q(G)\begin{array}{rcl}| 2\omega \left({e}_{3}\times \widehat{u}){| }_{{L}^{q}\left({\mathscr{F}})}& \le & 2\omega | \widehat{u}{| }_{{L}^{q}\left({\mathscr{F}})},\\ | {\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}})\widehat{h}{| }_{{H}^{1,q}\left({\mathscr{F}})}& \le & C{\omega }^{2}| y^{\prime} \cdot {\nu }_{{\mathscr{G}}}{| }_{{H}^{1,\infty }\left({\mathscr{F}})}| \widehat{h}{| }_{{H}^{1,q}\left({\mathscr{F}})}\le {C}_{q,{\mathscr{G}}}{\omega }^{2}| \widehat{h}{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}\end{array}for (u^,h^)∈H2,q(F)3×Bq,q3−1/q(G)\left(\widehat{u},\widehat{h})\in {H}^{2,q}{\left({\mathscr{F}})}^{3}\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), q∈(1,∞)q\in \left(1,\infty ), it follows from Proposition 2.1 and (4.5) that ℛXq({(τ∂τ)ℓFλR(λ):λ∈Σε,λ1})≤c0(2ωλ1−1+Cq,Gω2λ1−1){{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }{F}_{\lambda }{\mathscr{R}}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})\le {c}_{0}\left(2\omega {\lambda }_{1}^{-1}+{C}_{q,{\mathscr{G}}}{\omega }^{2}{\lambda }_{1}^{-1})for any λ1≥λ0{\lambda }_{1}\ge {\lambda }_{0}. We shall choose λ1{\lambda }_{1}large enough such that λ1≥4(Cq,G+1)c0max(1,ω2)≕C0max(1,ω2).{\lambda }_{1}\ge 4\left({C}_{q,{\mathscr{G}}}+1){c}_{0}\max \left(1,{\omega }^{2})\hspace{0.33em}=: \hspace{0.33em}{C}_{0}\max \left(1,{\omega }^{2}).Then, we have (4.7)ℛXq({(τ∂τ)ℓFλR(λ):λ∈Σε,λ1})≤12.{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }{F}_{\lambda }{\mathscr{R}}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})\le \frac{1}{2}.Hence, from (4.6) and (4.7), we obtain ∣FλR(λ)G∣Xq≤12∣FλG∣Xq.| {F}_{\lambda }R\left(\lambda )G{| }_{{{\mathcal{X}}}_{q}}\le \frac{1}{2}| {F}_{\lambda }G{| }_{{{\mathcal{X}}}_{q}}.Namely, we have FλR(λ)Fλ−1∈ℒ(Xq){F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1}\in {\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q})with ∣FλR(λ)Fλ−1∣ℒ(Xq)≤1/2| {F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1}{| }_{{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q})}\le 1\hspace{0.1em}\text{/}\hspace{0.1em}2. Then, the Neumann series argument implies the existence of the inverse (I−FλR(λ)Fλ−1)−1{\left(I-{F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1})}^{-1}of FλR(λ)Fλ−1{F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1}in ℒ(Xq){\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q}). Hence, the operator Fλ−1(I−FλR(λ)Fλ−1)−1Fλ=(I−R(λ))−1{F}_{\lambda }^{-1}{\left(I-{F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1})}^{-1}{F}_{\lambda }={\left(I-R\left(\lambda ))}^{-1}exists in ℒ(Xq){\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q})for each λ∈Σε,λ1\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}. Setting Uω(λ)≔U0(λ)(I−R(λ))−1,Qω(λ)≔Q0(λ)(I−R(λ))−1,ℋω(λ)≔ℋ0(λ)(I−R(λ))−1,{{\mathcal{U}}}_{\omega }\left(\lambda ):= {{\mathcal{U}}}_{0}\left(\lambda ){\left(I-R\left(\lambda ))}^{-1},\hspace{1.0em}{{\mathcal{Q}}}_{\omega }\left(\lambda ):= {{\mathcal{Q}}}_{0}\left(\lambda ){\left(I-R\left(\lambda ))}^{-1},\hspace{1.0em}{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ):= {{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda ){\left(I-R\left(\lambda ))}^{-1},we see that (Uω(λ),Qω(λ),ℋω(λ))FλG\left({{\mathcal{U}}}_{\omega }\left(\lambda ),{{\mathcal{Q}}}_{\omega }\left(\lambda ),{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )){F}_{\lambda }Gsolves (4.2) such that the estimates (4.3) are valid with c=4c0c=4{c}_{0}. Finally, the uniqueness of solutions to (4.2) follows from the duality argument. Suppose that u^∈H2,q(F)3\widehat{u}\in {H}^{2,q}{\left({\mathscr{F}})}^{3}, q^∈H1,q(F)\widehat{q}\in {H}^{1,q}\left({\mathscr{F}}), and h^∈Bq,q3−1/q(G)\widehat{h}\in {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfy problem (4.2) with (f^u,g^d,g^uτ,g^uv,f^h)({\widehat{f}}_{u},{\widehat{g}}_{d},{\widehat{g}}_{u\tau },{\widehat{g}}_{uv},{\widehat{f}}_{h})vanishing. For any λ∈Σε,λ1\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}, Φ∈Cc∞(F)3\Phi \in {C}_{c}^{\infty }{\left({\mathscr{F}})}^{3}, we consider that (4.8)λ¯v^−μΔv^+2(−ω)(e3×v^)+∇p^=Φ,in F,divv^=0,in F,PG(2μD(v^)νG)=0,on G,2μD(v^)νG⋅νG−p^+ℬGθ^=0,on G,λ¯θ^−(P0Gv^)⋅νG=0,on G,\left\{\begin{array}{ll}\overline{\lambda }\widehat{v}-\mu \Delta \widehat{v}+2\left(-\omega )\left({e}_{3}\times \widehat{v})+\nabla \widehat{p}=\Phi ,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{v}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{v}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{v}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{p}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \overline{\lambda }\widehat{\theta }-\left({P}_{0}^{{\mathscr{G}}}\widehat{v})\cdot {\nu }_{{\mathscr{G}}}=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where λ¯\overline{\lambda }stands for the complex conjugate of λ\lambda . Then, according to the aforementioned discussion, there exists a solution (v^,p^,θ^)∈H2,q(F)3×H1,q(F)×Bq,q2−1/q(G)\left(\widehat{v},\widehat{p},\widehat{\theta })\in {H}^{2,q}{\left({\mathscr{F}})}^{3}\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})of (4.8) with 1<q<∞1\lt q\lt \infty . Hence, the divergence theorem gives (u^,Φ)F=(u^,λ¯v^)F−(u^,−μΔv^+∇p^)F+(u^,2(−ω)(e3×v^))F=(u^,λ¯v^)F−(u^,(2μD(v^)−p^I)νG)G+2μ(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v^)F−(u^⋅νG,−ℬGθ^)G+2μ(D(u^),D(v^))F+(2ω(e3×u^),v^)F.\begin{array}{rcl}{\left(\widehat{u},\Phi )}_{{\mathscr{F}}}& =& {\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}-{\left(\widehat{u},-\mu \Delta \widehat{v}+\nabla \widehat{p})}_{{\mathscr{F}}}+{\left(\widehat{u},2\left(-\omega )\left({e}_{3}\times \widehat{v}))}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}-{\left(\widehat{u},\left(2\mu D\left(\widehat{v})-\widehat{p}I){\nu }_{{\mathscr{G}}})}_{{\mathscr{G}}}+2\mu {\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}-{\left(\widehat{u}\cdot {\nu }_{{\mathscr{G}}},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+2\mu {\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}.\end{array}Here, we have used the identity (e3×u^,v^)F=−(u^,e3×v^)F{\left({e}_{3}\times \widehat{u},\widehat{v})}_{{\mathscr{F}}}=-{\left(\widehat{u},{e}_{3}\times \widehat{v})}_{{\mathscr{F}}}. By (4.8)5{\left(4.8)}_{5}, it holds (u^⋅νG,−ℬGθ^)G=(λh^,−ℬGθ^)G+1∣F∣∫Fv^dy,−ℬGθ^G=(λh^,−ℬGθ^)G,{\left(\widehat{u}\cdot {\nu }_{{\mathscr{G}}},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}={\left(\lambda \widehat{h},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+{\left(\frac{1}{| {\mathscr{F}}| }\mathop{\int }\limits_{{\mathscr{F}}}\widehat{v}{\rm{d}}y,-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta }\right)}_{{\mathscr{G}}}={\left(\lambda \widehat{h},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}},since integrating (4.8)5{\left(4.8)}_{5}over G{\mathscr{G}}gives (1,θ^)G=0{\left(1,\widehat{\theta })}_{{\mathscr{G}}}=0due to λ≠0\lambda \ne 0. Noting the self-adjointness of ℬG{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}, we deduce that (u^,Φ)F=(u^,λ¯v^)F+(λℬGh^,θ^)G+2μ(D(u^),D(v^))F+(2ω(e3×u^),v^)F.{\left(\widehat{u},\Phi )}_{{\mathscr{F}}}={\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}+{\left(\lambda {{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h},\widehat{\theta })}_{{\mathscr{G}}}+2\mu {\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}.Analogously, we obtain 0=(λu^−μΔu^+2ω(e3×u^)+∇q^,v^)F=(u^,λ¯v)F−((2μD(u^)−p^I)νG,v^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(−ℬGh^,v^⋅νG)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(−ℬGh^,λ¯θ^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(λh^,−ℬGθ^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(u^⋅νG,−ℬGθ^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F.\begin{array}{rcl}0& =& {\left(\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q},\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(\left(2\mu D\left(\widehat{u})-\widehat{p}I){\nu }_{{\mathscr{G}}},\widehat{v})}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h},\widehat{v}\cdot {\nu }_{{\mathscr{G}}})}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h},\overline{\lambda }\widehat{\theta })}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(\lambda \widehat{h},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(\widehat{u}\cdot {\nu }_{{\mathscr{G}}},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}.\end{array}Hence, we arrive at (u^,Φ)F=0{\left(\widehat{u},\Phi )}_{{\mathscr{F}}}=0for any Φ∈Cc∞(F)3\Phi \in {C}_{c}^{\infty }{\left({\mathscr{F}})}^{3}. Since Φ\Phi is arbitrary, we obtain u^=0\widehat{u}=0in F{\mathscr{F}}. Then, it follows from the equation (4.2)5{\left(4.2)}_{5}that h^=0\widehat{h}=0on G{\mathscr{G}}, as λ≠0\lambda \ne 0. This completes the proof.□Using the operator valued Fourier multiplier theorem due to Prüss [21], we find that Theorem 4.1 immediately follows from Lemma 4.2, see also the discussion in the proof of [39, Thm. 3.5].5Decay estimate for the linearized equationsIn this section, we shall address some exponential decay property of the linearized system (4.1). To this end, let {φm}m=14{\left\{{\varphi }_{m}\right\}}_{m=1}^{4}be an orthogonal basis of N(ℬG)∪C{\mathsf{N}}\left({{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}})\cup {\mathbb{C}}with respect to L2{L}^{2}inner-product (⋅,⋅)G{\left(\cdot ,\cdot )}_{{\mathscr{G}}}, where N(ℬG){\mathsf{N}}\left({{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}})stands for the null space of ℬG{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}. Here, φm{\varphi }_{m}, m=1,2,3,4m=1,2,3,4, are given by φ1=∣G∣−1/2{\varphi }_{1}=| {\mathscr{G}}{| }^{-1\text{/}2}, and φℓ+1=Cℓyℓ{\varphi }_{\ell +1}={C}_{\ell }{y}_{\ell }, ℓ=1,2,3\ell =1,2,3, respectively, normalized by (φj,φk)G=δjk{\left({\varphi }_{j},{\varphi }_{k})}_{{\mathscr{G}}}={\delta }_{jk}, where Cℓ{C}_{\ell }are constants. Assume that λ1{\lambda }_{1}is the same constant as in Lemma 4.2 in what follows. This section is dedicated to show the following theorem.Theorem 5.1Let 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Set J=(0,T)J=\left(0,T)with T>0T\gt 0. Then, there exists a constant ε0>0{\varepsilon }_{0}\gt 0such that the following assertion is valid: Let u0∈Bq,p2(δ−1/p)(F)3{u}_{0}\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}, h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), and (fu,gd,guτ,guv,fh)∈Fδ(J;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})that satisfy the compatibility conditions given in Theorem 4.1. The problem (4.1) admits a unique solution (u,q,TrG[q],h)∈Eδ(J;F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})possessing the estimate∣eε0t(u,q,TrG[q],h)∣Eδ(J;F)≤C∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)+∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p\begin{array}{l}| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left(J;{\mathscr{F}})}\\ \hspace{1.0em}\le C\left[| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right.\\ \hspace{1.0em}\hspace{1.0em}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}\left.+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right]\end{array}with some constant C independent of T.We first decompose (u,q,h)\left(u,q,h)of (4.1). To this end, we recall a unique solvability result of the weak Dirichlet problem: For any f∈Lq(F)3f\in {L}^{q}{\left({\mathscr{F}})}^{3}, 1<q<∞1\lt q\lt \infty , there exists a unique θ∈H˙01,q(F)\theta \in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}})satisfying (5.1)(∇θ,∇φ)F=(f,∇φ)Ffor any φ∈H˙01,q′(F){\left(\nabla \theta ,\nabla \varphi )}_{{\mathscr{F}}}={(f,\nabla \varphi )}_{{\mathscr{F}}}\hspace{1.0em}\hspace{0.1em}\text{for any\hspace{0.5em}}\hspace{0.1em}\varphi \in {\dot{H}}_{0}^{1,q^{\prime} }\hspace{0.33em}\left({\mathscr{F}})\text{}and possessing the estimate ∣∇θ∣Lq(F)≤C∣f∣Lq(F)| \nabla \theta {| }_{{L}^{q}\left({\mathscr{F}})}\le C| f{| }_{{L}^{q}\left({\mathscr{F}})}with a constant CCindependent of the choices of θ\theta , φ\varphi , and ff. Then, for any f∈Lq(F)3f\in {L}^{q}{\left({\mathscr{F}})}^{3}, we define the operators PF{{\mathbb{P}}}_{{\mathscr{F}}}and QF{{\mathbb{Q}}}_{{\mathscr{F}}}by ∇QFf≔∇θ\nabla {{\mathbb{Q}}}_{{\mathscr{F}}}f:= \nabla \theta and PFf≔(I−∇QF)f∈Jq(F){{\mathbb{P}}}_{{\mathscr{F}}}f:= \left(I-\nabla {{\mathbb{Q}}}_{{\mathscr{F}}})f\in {J}_{q}\left({\mathscr{F}})with θ∈H˙01,q(F)\theta \in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}})satisfying (5.1). Here, Jq(F){J}_{q}\left({\mathscr{F}})is the solenoidal space given by Jq(F)≔{f∈Lq(F)3:(f,∇φ)F=0for all φ∈H˙01,q′(F)}.{J}_{q}\left({\mathscr{F}}):= \{f\in {L}^{q}{\left({\mathscr{F}})}^{3}\hspace{0.33em}:\hspace{0.33em}{(f,\nabla \varphi )}_{{\mathscr{F}}}=0\hspace{1em}\hspace{0.1em}\text{for all\hspace{0.5em}}\hspace{0.1em}\varphi \in {\dot{H}}_{0}^{1,q^{\prime} }\hspace{0.33em}\left({\mathscr{F}})\text{}\}.Next, we consider the following systems: (5.2)∂tv∗+2λ2v∗−μΔv∗+2ω(e3×v∗)+∇π∗=fu,in F,divv∗=gd,in F,PG(2μD(v∗)νG)=guτ,on G,2μD(v∗)νG⋅νG−π∗+ℬGη∗=guv,on G,∂tη∗+2λ2η∗−(P0Gu∗)⋅νG=fh,on G,v∗(0)=u0,in F,η∗(0)=h0,on G.\left\{\begin{array}{ll}{\partial }_{t}{v}_{\ast }+2{\lambda }_{2}{v}_{\ast }-\mu \Delta {v}_{\ast }+2\omega \left({e}_{3}\times {v}_{\ast })+\nabla {\pi }_{\ast }={f}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{v}_{\ast }={g}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({v}_{\ast }){\nu }_{{\mathscr{G}}})={g}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({v}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{\pi }_{\ast }+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}{\eta }_{\ast }={g}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{\eta }_{\ast }+2{\lambda }_{2}{\eta }_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{u}_{\ast })\cdot {\nu }_{{\mathscr{G}}}={f}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {v}_{\ast }\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\eta }_{\ast }\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.(5.3)∂tu∗−μΔu∗+2ω(e3×u∗)+∇q∗=2λ2PFv˜∗,in F,divu∗=0,in F,PG(2μD(u∗)νG)=0,on G,2μD(u∗)νG⋅νG−q∗+ℬGh∗=0,on G,∂th∗−(P0Gu∗)⋅νG=2λ2η˜∗,on G,u∗(0)=0,in F,h∗(0)=0,on G.\left\{\begin{array}{ll}{\partial }_{t}{u}_{\ast }-\mu \Delta {u}_{\ast }+2\omega \left({e}_{3}\times {u}_{\ast })+\nabla {q}_{\ast }=2{\lambda }_{2}{{\mathbb{P}}}_{{\mathscr{F}}}{\widetilde{v}}_{\ast },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{u}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({u}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({u}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{q}_{\ast }+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}{h}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{h}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{u}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=2{\lambda }_{2}{\widetilde{\eta }}_{\ast },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {u}_{\ast }\left(0)=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {h}_{\ast }\left(0)=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Here, we have set v˜∗≔v∗−(v∗,1)F−∑α=1,2(v∗,eα×y)F−ω∫Gη˜∗yαy3dy(eα×y)−(v∗,e3×y)F(e3×y){\widetilde{v}}_{\ast }:= {v}_{\ast }-{\left({v}_{\ast },1)}_{{\mathscr{F}}}-\sum _{\alpha =1,2}\left({\left({v}_{\ast },{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}{\widetilde{\eta }}_{\ast }{y}_{\alpha }{y}_{3}{\rm{d}}y\right)\left({e}_{\alpha }\times y)-{\left({v}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}\left({e}_{3}\times y)and η˜∗≔η∗−∑m=14(η∗,φm)Gφm{\widetilde{\eta }}_{\ast }:= {\eta }_{\ast }-{\sum }_{m=1}^{4}{\left({\eta }_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}{\varphi }_{m}, respectively. Now, setting (5.4)u≔v∗+u∗+2λ2∫0t(v∗(s),1)F+∑α=1,2(v∗(s),eα×y)F−ω∫Gη˜∗(s)yαy3dy(eα×y)+(v∗(s),e3×y)F(e3×y)ds,u:= {v}_{\ast }+{u}_{\ast }+2{\lambda }_{2}\underset{0}{\overset{t}{\int }}\left\{{\left({v}_{\ast }\left(s),1)}_{{\mathscr{F}}}+\sum _{\alpha =1,2}\left({\left({v}_{\ast }\left(s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}{\widetilde{\eta }}_{\ast }\left(s){y}_{\alpha }{y}_{3}{\rm{d}}y\right)\left({e}_{\alpha }\times y)+{\left({v}_{\ast }\left(s),{e}_{3}\times y)}_{{\mathscr{F}}}\left({e}_{3}\times y)\right\}{\rm{d}}s,and (5.5)q≔π∗+q∗−2λ2∇QFv˜∗,h≔η∗+h∗+2λ2∑m=14∫0t(η∗(s),φm)Gφmds,q:= {\pi }_{\ast }+{q}_{\ast }-2{\lambda }_{2}\nabla {{\mathbb{Q}}}_{{\mathscr{F}}}{\widetilde{v}}_{\ast },\hspace{1.0em}h:= {\eta }_{\ast }+{h}_{\ast }+2{\lambda }_{2}\mathop{\sum }\limits_{m=1}^{4}\underset{0}{\overset{t}{\int }}{\left({\eta }_{\ast }\left(s),{\varphi }_{m})}_{{\mathscr{G}}}{\varphi }_{m}{\rm{d}}s,we see that (u,q,h)\left(u,q,h)is a solution to (4.1). In the following, we construct the solutions of (5.2) and (5.3), respectively.Step 1: Study of (5.2). For the shifted system (5.2), we have the next theorem.Theorem 5.2Assume that 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Set J=(0,T)J=\left(0,T)with T>0T\gt 0. Let u0∈Bq,p2(δ−1/p)(F)3{u}_{0}\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}, h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), and (fu,gd,guτ,guv,fh)∈Fδ(J;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})that satisfy the compatibility conditions given in Theorem 4.1. Then, for any λ2>λ1/2{\lambda }_{2}\gt {\lambda }_{1}\hspace{0.1em}\text{/}\hspace{0.1em}2, system (5.2) has a unique solution (v∗,π∗,TrG[π∗],η∗)∈Eδ(J;F)\left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{G}}}\left[{\pi }_{\ast }],{\eta }_{\ast })\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})possessing the estimate∣(v∗,π∗,TrF[π∗],η∗)∣Eδ(J;F)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣(fu,gd,guτ,guv,fh)∣Fδ(J;F))| \left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{F}}}\left[{\pi }_{\ast }],{\eta }_{\ast }){| }_{{{\mathbb{E}}}_{\delta }\left(J;{\mathscr{F}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| ({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})for some constant C independent of TT, ω\omega , λ1{\lambda }_{1}, and λ2{\lambda }_{2}.ProofFor λ∈Σε,λ1\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}, let Uω(λ){{\mathcal{U}}}_{\omega }\left(\lambda ), Qω(λ){{\mathcal{Q}}}_{\omega }\left(\lambda ), and ℋω(λ){{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )be operators given in Lemma 4.2. Let λ2{\lambda }_{2}and ε1{\varepsilon }_{1}be numbers for which 2λ2−λ1>ε1>02{\lambda }_{2}-{\lambda }_{1}\gt {\varepsilon }_{1}\gt 0. Then, for every λ∈−ε1+Σε,0\lambda \in -{\varepsilon }_{1}+{\Sigma }_{\varepsilon ,0}, it holds λ+2λ2∈λ1+Σε,0\lambda +2{\lambda }_{2}\in {\lambda }_{1}+{\Sigma }_{\varepsilon ,0}. Hence, by Lemma 4.2, we obtain ℛXq→H2−j,q(F)3({(τ∂τ)ℓ(λj/2Uω(λ+2λ2)):−ε1+λ∈Σε,0})≤c,ℛXq→Lq(F)3({(τ∂τ)ℓ∇Qω(λ+2λ2):−ε1+λ∈Σε,0})≤c,ℛXq→Bq,q3−1/q−k(G)({(τ∂τ)ℓ(λkℋω(λ+2λ2)):−ε1+λ∈Σε,0})≤c\begin{array}{rcl}{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {H}^{2-j,q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{j\text{/}2}{{\mathcal{U}}}_{\omega }\left(\lambda +2{\lambda }_{2}))\hspace{0.33em}:\hspace{0.33em}-{\varepsilon }_{1}+\lambda \in {\Sigma }_{\varepsilon ,0}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {L}^{q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\nabla {{\mathcal{Q}}}_{\omega }\left(\lambda +2{\lambda }_{2})\hspace{0.33em}:\hspace{0.33em}-{\varepsilon }_{1}+\lambda \in {\Sigma }_{\varepsilon ,0}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {B}_{q,q}^{3-1\text{/}q-k}\left({\mathscr{G}})}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{k}{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda +2{\lambda }_{2}))\hspace{0.33em}:\hspace{0.33em}-{\varepsilon }_{1}+\lambda \in {\Sigma }_{\varepsilon ,0}\})& \le & c\end{array}for ℓ=0,1\ell =0,1, j=0,1,2j=0,1,2, and k=0,1k=0,1. Employing the argument in the proof of [39, Thm. 3.5] readily implies the required assertion. This completes the proof.□For every ε0>0{\varepsilon }_{0}\gt 0, we see that eε0t(v∗,π∗,η∗){e}^{{\varepsilon }_{0}t}\left({v}_{\ast },{\pi }_{\ast },{\eta }_{\ast })satisfies ∂t(eε0tv∗)+(2λ2−ε0)(eε0v∗)−μΔ(eε0tv∗)+2ω(e3×(eε0tv∗))+∇(eε0tπ∗)=eε0tfu,in F,div(eε0tv∗)=eε0tgd,in F,PG(2μD(eε0tv∗)νG)=eε0tguτ,on G,2μD(eε0tv∗)νG⋅νG−(eε0tπ∗)+ℬG(eε0tη∗)=eε0tguv,on G,∂t(eε0tη∗)+(2λ2−ε0)(eε0tη∗)−(P0G(eε0tv∗))⋅νG=eε0tfh,on G,v∗(0)=u0,in F,η∗(0)=h0,on G.\left\{\begin{array}{ll}{\partial }_{t}\left({e}^{{\varepsilon }_{0}t}{v}_{\ast })+\left(2{\lambda }_{2}-{\varepsilon }_{0})\left({e}^{{\varepsilon }_{0}}{v}_{\ast })-\mu \Delta \left({e}^{{\varepsilon }_{0}t}{v}_{\ast })+2\omega \left({e}_{3}\times \left({e}^{{\varepsilon }_{0}t}{v}_{\ast }))+\nabla \left({e}^{{\varepsilon }_{0}t}{\pi }_{\ast })={e}^{{\varepsilon }_{0}t}{f}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\left({e}^{{\varepsilon }_{0}t}{v}_{\ast })={e}^{{\varepsilon }_{0}t}{g}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({e}^{{\varepsilon }_{0}t}{v}_{\ast }){\nu }_{{\mathscr{G}}})={e}^{{\varepsilon }_{0}t}{g}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({e}^{{\varepsilon }_{0}t}{v}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\left({e}^{{\varepsilon }_{0}t}{\pi }_{\ast })+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\left({e}^{{\varepsilon }_{0}t}{\eta }_{\ast })={e}^{{\varepsilon }_{0}t}{g}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}\left({e}^{{\varepsilon }_{0}t}{\eta }_{\ast })+\left(2{\lambda }_{2}-{\varepsilon }_{0})\left({e}^{{\varepsilon }_{0}t}{\eta }_{\ast })-\left({P}_{0}^{{\mathscr{G}}}\left({e}^{{\varepsilon }_{0}t}{v}_{\ast }))\cdot {\nu }_{{\mathscr{G}}}={e}^{{\varepsilon }_{0}t}{f}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {v}_{\ast }\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\eta }_{\ast }\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Given ε0>0{\varepsilon }_{0}\gt 0, we choose λ2>0{\lambda }_{2}\gt 0suitably large such that 2λ2−λ1>ε0>02{\lambda }_{2}-{\lambda }_{1}\gt {\varepsilon }_{0}\gt 0. Then, from Theorem 5.2, we obtain the following corollary, which gives the decay property of solutions to (5.2).Corollary 5.3Let 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Set J=(0,T)J=\left(0,T)with T>0T\gt 0. Let u0∈Bq,p2(δ−1/p)(F)3{u}_{0}\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}, h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), and (fu,gd,guτ,guv,fh)∈Fδ(J;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})that satisfy the compatibility conditions given in Theorem 4.1. There exists a constant λ2>0{\lambda }_{2}\gt 0such that the system (5.2) admits a unique solution (v∗,π∗,TrG[π∗],η∗)∈Eδ(J;F)\left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{G}}}\left[{\pi }_{\ast }],{\eta }_{\ast })\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})possessing the estimate∣eε0t(v∗,π∗,TrG[π∗],η∗)∣Eδ(J;F)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F))| {e}^{{\varepsilon }_{0}t}\left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{G}}}\left[{\pi }_{\ast }],{\eta }_{\ast }){| }_{{{\mathbb{E}}}_{\delta }\left(J;{\mathscr{F}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})with a constant C independent of TT, ω\omega , λ1{\lambda }_{1}, and λ2{\lambda }_{2}, where ε0{\varepsilon }_{0}is a number such that 0<ε0<2λ2−λ10\lt {\varepsilon }_{0}\lt 2{\lambda }_{2}-{\lambda }_{1}.Step 2: Study of (5.3). To study (5.3), we rely on the following decomposition: (5.6)u∗(y,t)=U∗(y,t)+d3[h∗](e3×y),q∗(y,t)=Q∗(y,t)+ωd3[h∗]∣y′∣2+1∣G∣∫GCGh∗dG,{u}_{\ast }(y,t)={U}_{\ast }(y,t)+{d}_{3}\left[{h}_{\ast }]\left({e}_{3}\times y),\hspace{1.0em}{q}_{\ast }(y,t)={Q}_{\ast }(y,t)+\omega {d}_{3}\left[{h}_{\ast }]| y^{\prime} {| }^{2}+\frac{1}{| {\mathscr{G}}| }\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }{\rm{d}}{\mathscr{G}},where we have set dℓ[h∗]≔−ωSℓ∫Gh∗(y,t)(e3×y)⋅(eℓ×y)dG,Sℓ≔∣eℓ×y∣L2(F)2=∫F(∣y∣2−yℓ2)dy,CGh∗≔ℬGh∗−ωd3[h∗]∣y′∣2\begin{array}{rcl}{d}_{\ell }\left[{h}_{\ast }]& := & -\frac{\omega }{{{\mathcal{S}}}_{\ell }}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }(y,t)\left({e}_{3}\times y)\cdot \left({e}_{\ell }\times y){\rm{d}}{\mathscr{G}},\\ {{\mathcal{S}}}_{\ell }& := & | {e}_{\ell }\times y{| }_{{L}^{2}\left({\mathscr{F}})}^{2}=\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left(| y{| }^{2}-{y}_{\ell }^{2}){\rm{d}}y,\\ {{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }& := & {{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}{h}_{\ast }-\omega {d}_{3}\left[{h}_{\ast }]| y^{\prime} {| }^{2}\end{array}with ℓ=1,2,3\ell =1,2,3. We further set C^Gh∗≔CGh∗−1∣G∣∫GCGh∗dG.{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }:= {{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }-\frac{1}{| {\mathscr{G}}| }\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }{\rm{d}}{\mathscr{G}}.Noting that 2e3×(e3×y)=−(2y1,2y2,0)=−∇∣y′∣2,∂td3[h∗]=−ωS3∫G∂th∗∣y′∣2dG=−ωS3∫G((P0GU∗)⋅νG+2λ2η˜∗)∣y′∣2dG,\begin{array}{rcl}2{e}_{3}\times \left({e}_{3}\times y)& =& -\left(2{y}_{1},2{y}_{2},0)=-\nabla | y^{\prime} {| }^{2},\\ {\partial }_{t}{d}_{3}\left[{h}_{\ast }]& =& -\frac{\omega }{{{\mathcal{S}}}_{3}}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\partial }_{t}{h}_{\ast }| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}=-\frac{\omega }{{{\mathcal{S}}}_{3}}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}(\left({P}_{0}^{{\mathscr{G}}}{U}_{\ast })\cdot {\nu }_{{\mathscr{G}}}+2{\lambda }_{2}{\widetilde{\eta }}_{\ast })| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}},\end{array}the system (5.3) can be rewritten as follows: (5.7)∂tU∗−Lω,yU∗+∇Q∗=2λ2f˜w,in F,divU∗=0,in F,PG(2μD(U∗)νG)=0,on G,2μD(U∗)νG⋅νG−Q∗+C^Gh∗=0,on G,∂th∗−(P0GU∗)⋅νG=2λ2η˜∗,on G,U∗(0)=0,in F,h∗(0)=0,on G,\left\{\begin{array}{ll}{\partial }_{t}{U}_{\ast }-{L}_{\omega ,y}{U}_{\ast }+\nabla {Q}_{\ast }=2{\lambda }_{2}{\widetilde{f}}_{w},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{U}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({U}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{Q}_{\ast }+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{h}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{U}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=2{\lambda }_{2}{\widetilde{\eta }}_{\ast },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {U}_{\ast }\left(0)=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {h}_{\ast }\left(0)=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where we have set Lω,yw≔2ωS3∫F(P0Gw)⋅y′dy(e3×y)+μΔw−2ω(e3×w),f˜w≔PFv˜∗+ωS3∫Gη˜∗∣y′∣2dG(e3×y).\begin{array}{rcl}{L}_{\omega ,y}w& := & \frac{2\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({P}_{0}^{{\mathscr{G}}}w)\cdot y^{\prime} {\rm{d}}y\right)\left({e}_{3}\times y)+\mu \Delta w-2\omega \left({e}_{3}\times w),\\ {\widetilde{f}}_{w}& := & {{\mathbb{P}}}_{{\mathscr{F}}}{\widetilde{v}}_{\ast }+\frac{\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widetilde{\eta }}_{\ast }| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)\left({e}_{3}\times y).\end{array}Notice that it holds (e3×y)⋅νG=0\left({e}_{3}\times y)\cdot {\nu }_{{\mathscr{G}}}=0on G{\mathscr{G}}due to the axisymmetry of G{\mathscr{G}}.As a base space for our analysis, we use X0=Jq(F)×Bq,q2−1/q(G),{X}_{0}={J}_{q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}),and we set X¯1≔H2,q(F)3×Bq,q3−1/q(G).{\overline{X}}_{1}:= {H}^{2,q}{\left({\mathscr{F}})}^{3}\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}).Define a closed linear operator in X0{X}_{0}by means of Aq(U∗,h∗)≔(−Lω,yU∗+∇K(U∗,h∗),−(P0GU∗)⋅νG){{\mathcal{A}}}_{q}\left({U}_{\ast },{h}_{\ast }):= \left(-{L}_{\omega ,y}{U}_{\ast }+\nabla K\left({U}_{\ast },{h}_{\ast }),-\left({P}_{0}^{{\mathscr{G}}}{U}_{\ast })\cdot {\nu }_{{\mathscr{G}}})with domain X1≔D(Aq)⊂X¯1{X}_{1}:= {\mathsf{D}}\left({{\mathcal{A}}}_{q})\subset {\overline{X}}_{1}defined by D(Aq)≔{(U∗,h∗)∈X0∩X¯1:PG(2μD(U∗)νG)=0on G}.{\mathsf{D}}\left({{\mathcal{A}}}_{q}):= \left\{\left({U}_{\ast },{h}_{\ast })\in {X}_{0}\cap {\overline{X}}_{1}\hspace{0.33em}:\hspace{0.33em}{{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})=0\hspace{0.33em}\hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\right\}.Here, K(U∗,h∗)∈H˙01,q(F)K\left({U}_{\ast },{h}_{\ast })\in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}})is a functional that is a unique solution to the weak Dirichlet problem (5.8)(∇K(U∗,h∗),∇φ)F=(Lω,yU∗,∇φ)Ffor any φ∈H˙01,q′(F),K(U∗,h∗)=2μ(D(U∗)νG)⋅νG+C^Gh∗on G\left\{\begin{array}{ll}{\left(\nabla K\left({U}_{\ast },{h}_{\ast }),\nabla \varphi )}_{{\mathscr{F}}}={\left({L}_{\omega ,y}{U}_{\ast },\nabla \varphi )}_{{\mathscr{F}}}& \hspace{0.1em}\text{for any\hspace{0.5em}}\hspace{0.1em}\varphi \in {\dot{H}}_{0}^{1,q^{\prime} }\left({\mathscr{F}})\text{},\\ K\left({U}_{\ast },{h}_{\ast })=2\mu \left(D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})\cdot {\nu }_{{\mathscr{G}}}+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for (U∗,h∗)∈X0∩X¯1\left({U}_{\ast },{h}_{\ast })\in {X}_{0}\cap {\overline{X}}_{1}. Since (U∗,h∗)\left({U}_{\ast },{h}_{\ast })belongs to X0∩X¯1{X}_{0}\cap {\overline{X}}_{1}, we have ∣Lω,yU∗∣Lq(F)≤Cq,F(μ∣U∗∣H2,q(F)+ω∣U∗∣Lq(F)),∣2μ(D(U∗)νG)⋅νG+C^Gh∗∣Bq,q1−1/q(G)≤Cq,G(μ∣U∗∣H2,q(F)+σ∣h∗∣Bq,q3−1/q(G)+ω2∣h∗∣Bq,q2−1/q(G)),\begin{array}{rcl}| {L}_{\omega ,y}{U}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}& \le & {C}_{q,{\mathscr{F}}}(\mu | {U}_{\ast }{| }_{{H}^{2,q}\left({\mathscr{F}})}+\omega | {U}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}),\\ | 2\mu \left(D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})\cdot {\nu }_{{\mathscr{G}}}+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }{| }_{{B}_{q,q}^{1-1\text{/}q}\left({\mathscr{G}})}& \le & {C}_{q,{\mathscr{G}}}(\mu | {U}_{\ast }{| }_{{H}^{2,q}\left({\mathscr{F}})}+\sigma | {h}_{\ast }{| }_{{B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}})}+{\omega }^{2}| {h}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}),\end{array}which yields the estimate for ∇K(U∗,h∗)\nabla K\left({U}_{\ast },{h}_{\ast }): ∣∇K(U∗,h∗)∣Lq(F)≤Cq,G(μ∣U∗∣H2,q(F)+ω∣U∗∣Lq(F)+σ∣h∗∣Bq,q3−1/q(G)+ω2∣h∗∣Bq,q2−1/q(G)).| \nabla K\left({U}_{\ast },{h}_{\ast }){| }_{{L}^{q}\left({\mathscr{F}})}\le {C}_{q,{\mathscr{G}}}(\mu | {U}_{\ast }{| }_{{H}^{2,q}\left({\mathscr{F}})}+\omega | {U}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+\sigma | {h}_{\ast }{| }_{{B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}})}+{\omega }^{2}| {h}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}).As F{\mathscr{F}}is a bounded smooth domain, the weak Dirichlet problem (5.8) admits a unique solution K(U∗,h∗)∈H˙01,q(F)K\left({U}_{\ast },{h}_{\ast })\in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}}), i.e., the functional K(U∗,h∗)K\left({U}_{\ast },{h}_{\ast })is well defined. Notice that the solution K(U∗,h∗)K\left({U}_{\ast },{h}_{\ast })to (5.8) depends on ω\omega . Applying the similar argument in [25, Sec. 9.2.1], we see that the system (5.7) is equivalent to the abstract evolution equation z˙+Aqz=(2λ2f˜w,2λ2η˜∗),t>0,z(0)=0\dot{z}+{{\mathcal{A}}}_{q}z\left=(2{\lambda }_{2}{\widetilde{f}}_{w}\left,2{\lambda }_{2}{\widetilde{\eta }}_{\ast }),\hspace{1.0em}t\gt 0,\hspace{1.0em}z\left(0)=0with z=(U∗,h∗)z=\left({U}_{\ast },{h}_{\ast }). Employing the standard Neumann series argument, we can prove that the operator −Aq-{{\mathcal{A}}}_{q}generates an analytic C0{C}_{0}-semigroup in X0{X}_{0}.Lemma 5.4Let 1<q<∞1\lt q\lt \infty . Then −Aq-{{\mathcal{A}}}_{q}with domain D(Aq){\mathsf{D}}\left({{\mathcal{A}}}_{q})generates an analytic C0{C}_{0}-semigroup in X0{X}_{0}.ProofLet us consider the following resolvent problem: (5.9)λU^∗−Lω,yU^∗+∇Q^∗=f^u,in F,divU^∗=0,in F,PG(2μD(U^∗)νG)=0,on G,2μD(U^∗)νG⋅νG−Q^∗+C^Gh^∗=g^uv,on G,λh^∗−(P0GU^∗)⋅νG=f^h,on G\left\{\begin{array}{ll}\lambda {\widehat{U}}_{\ast }-{L}_{\omega ,y}{\widehat{U}}_{\ast }+\nabla {\widehat{Q}}_{\ast }={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{\widehat{U}}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{\widehat{Q}}_{\ast }+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast }={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for any λ∈Σε,λ4\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{4}}and (f^u,0,0,g^uv,f^h)∈Xq,σ({\widehat{f}}_{u},0,0,{\widehat{g}}_{uv},{\widehat{f}}_{h})\in {X}_{q,\sigma }with some positive constant λ4{\lambda }_{4}, where we have set Xq,σ≔Xq∩(Jq(F)×DIq(F)×H1,q(F)2×H1,q(F)×Bq,q2−1/q(G)).{X}_{q,\sigma }:= {X}_{q}\cap ({J}_{q}\left({\mathscr{F}})\times {{\rm{DI}}}_{q}\left({\mathscr{F}})\times {H}^{1,q}{\left({\mathscr{F}})}^{2}\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})).To find solutions of the aforementioned problem, let us consider (5.10)λu^−μΔu^+2ω(e3×u^)+∇q^=f^u,in F,divu^=0,in F,PG(2μD(u^)νG)=0,on G,2μD(u^)νG⋅νG−q^+ℬGh^=g^uv,on G,λh^−(P0Gu^)⋅νG=f^h,on G\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q}={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h}={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for (f^u,0,0,g^uv,f^h)∈Xq,σ({\widehat{f}}_{u},0,0,{\widehat{g}}_{uv},{\widehat{f}}_{h})\in {X}_{q,\sigma }and λ∈Σε,λ2\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{2}}. According to Lemma 4.2, there exists a unique solution (u^,q^,h^)∈(H2,q(F)3∩Jq(F))×H1,q(F)×Bq,q3−1/q(G)\left(\widehat{u},\widehat{q},\widehat{h})\in \left({H}^{2,q}{\left({\mathscr{F}})}^{3}\cap {J}_{q}\left({\mathscr{F}}))\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})of (5.10). Let Uω{{\mathcal{U}}}_{\omega }and ℋω{{\mathcal{ {\mathcal H} }}}_{\omega }be the operators given in Lemma 4.2. If we set q˜≔q^−∣G∣−1∫GCGh^dG\widetilde{q}:= \widehat{q}-| {\mathscr{G}}{| }^{-1}{\int }_{{\mathscr{G}}}{{\mathcal{C}}}_{{\mathscr{G}}}\widehat{h}{\rm{d}}{\mathscr{G}}, for λ∈Σε,λ2\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{2}}, we see that (u^,q˜,h^)\left(\widehat{u},\widetilde{q},\widehat{h})satisfies λu^−Lω,yu^+∇q˜=f^u+2ωS3∫F(P0GUω(λ)F0)⋅y′dy(e3×y),in F,divu^=0,in F,PG(2μD(u^)νG)=0,on G,2μD(u^)νG⋅νG−q˜+C^Gh^=f^uv+ωd3[ℋω(λ)F0]∣y′∣2,on G,λh^−(P0Gu^)⋅νG=f^h,on G,\left\{\begin{array}{ll}\lambda \widehat{u}-{L}_{\omega ,y}\widehat{u}+\nabla \widetilde{q}={\widehat{f}}_{u}+\frac{2\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({P}_{0}^{{\mathscr{G}}}{{\mathcal{U}}}_{\omega }\left(\lambda ){F}_{0})\cdot y^{\prime} {\rm{d}}y\right)\left({e}_{3}\times y),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widetilde{q}+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widehat{h}={\widehat{f}}_{uv}+\omega {d}_{3}\left[{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ){F}_{0}]| y^{\prime} {| }^{2},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where we have set F0=(f^u,0,⋯,0,λ1/2g^uv,g^uv,f^h)∈Xq{F}_{0}=({\widehat{f}}_{u},0,\cdots \hspace{0.33em},0,{\lambda }^{1\text{/}2}{\widehat{g}}_{uv},{\widehat{g}}_{uv},{\widehat{f}}_{h})\in {{\mathcal{X}}}_{q}. Since it holds 2ωS3∫F(P0GUω(λ)F0)⋅y′dy(e3×y)Lq(F)≤C1ωλ3−1∣F0∣Xq,∣ωd3[ℋω(λ)F0]∣y′∣2∣Bq,q2−1/q(G)≤C1ω2λ3−1∣F0∣Xq\begin{array}{rcl}{\left|\frac{2\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({P}_{0}^{{\mathscr{G}}}{{\mathcal{U}}}_{\omega }\left(\lambda ){F}_{0})\cdot y^{\prime} {\rm{d}}y\right)\left({e}_{3}\times y)\right|}_{{L}^{q}\left({\mathscr{F}})}& \le & {C}_{1}\omega {\lambda }_{3}^{-1}| {F}_{0}{| }_{{{\mathcal{X}}}_{q}},\\ | \omega {d}_{3}\left[{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ){F}_{0}]| y^{\prime} {| }^{2}{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}& \le & {C}_{1}{\omega }^{2}{\lambda }_{3}^{-1}| {F}_{0}{| }_{{{\mathcal{X}}}_{q}}\end{array}for any λ3≥λ2{\lambda }_{3}\ge {\lambda }_{2}, we shall take λ3{\lambda }_{3}sufficiently large such that λ3≥4C1max(ω2,1){\lambda }_{3}\ge 4{C}_{1}\max \left({\omega }^{2},1)so that it follows from the Neumann series argument that (5.9) admits a unique solution (U^∗,Q^∗,h^∗)∈(H2,q(F)3∩Jq(F))×H1,q(F)×Bq,q3−1/q(G)\left({\widehat{U}}_{\ast },{\widehat{Q}}_{\ast },{\widehat{h}}_{\ast })\in \left({H}^{2,q}{\left({\mathscr{F}})}^{3}\cap {J}_{q}\left({\mathscr{F}}))\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})with λ4≥λ3{\lambda }_{4}\ge {\lambda }_{3}, where the resolvent estimate ∣(∣λ∣U^∗,∣λ∣1/2∇U^∗,∇2U^∗)∣Lq(F)+∣∇Q^∗∣Lq(F)+∣(∣λ∣h^∗,∇h^∗)∣Bq,q2−1/q(G)≤C(∣f^u∣Lq(F)+∣λ∣1/2∣f^uv∣Lq(F)+∣f^uv∣H1,q(F)+∣f^h∣Bq,q2−1/q(G))| \left(| \lambda | {\widehat{U}}_{\ast },| \lambda {| }^{1\text{/}2}\nabla {\widehat{U}}_{\ast },{\nabla }^{2}{\widehat{U}}_{\ast }){| }_{{L}^{q}\left({\mathscr{F}})}+| \nabla {\widehat{Q}}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+| \left(| \lambda | {\widehat{h}}_{\ast },\nabla {\widehat{h}}_{\ast }){| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}\le C(| {\widehat{f}}_{u}{| }_{{L}^{q}\left({\mathscr{F}})}+| \lambda {| }^{1\text{/}2}| {\widehat{f}}_{uv}{| }_{{L}^{q}\left({\mathscr{F}})}+| {\widehat{f}}_{uv}{| }_{{H}^{1,q}\left({\mathscr{F}})}+| {\widehat{f}}_{h}{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})})is valid with a constant CCdepending only on qq, G{\mathscr{G}}, μ\mu , and σ\sigma . If we set f^uv≡0{\widehat{f}}_{uv}\equiv 0, the resolvent problem λz^+Aqz^=f^with z^=(U^∗,h^∗)  and f^=(f^u,f^h)\lambda \widehat{z}+{{\mathcal{A}}}_{q}\widehat{z}=\widehat{f}\hspace{1.0em}\hspace{0.1em}\text{with\hspace{0.5em}}\hspace{0.1em}\widehat{z}=\left({\widehat{U}}_{\ast },{\widehat{h}}_{\ast })\hspace{0.1em}\text{\hspace{0.5em} and\hspace{0.5em}}\hspace{0.1em}\widehat{f}=({\widehat{f}}_{u},{\widehat{f}}_{h})\text{}is equivalent to (5.9) with Q^∗=K(U^∗,h^∗){\widehat{Q}}_{\ast }=K\left({\widehat{U}}_{\ast },{\widehat{h}}_{\ast })and admits a unique solution z^∈X1\widehat{z}\in {X}_{1}satisfying the resolvent estimate ∣λ∣∣z^∣X0+∣z^∣X¯1≤C∣f^∣X0,λ∈Σε,λ4.| \lambda | | \widehat{z}{| }_{{X}_{0}}+| \widehat{z}{| }_{{\overline{X}}_{1}}\le C| \widehat{f}{| }_{{X}_{0}},\hspace{1.0em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{4}}.Finally, the closedness of Aq{{\mathcal{A}}}_{q}follows from the fact that the resolvent set is not empty. This completes the proof.□To show the exponential decay property of the system (5.7), we introduce the following functional spaces: Let B˜q,q2−1/q(G){\widetilde{B}}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})be the set of all g∈Bq,q2−1/q(G)g\in {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfying (g,φm)G=0{\left(g,{\varphi }_{m})}_{{\mathscr{G}}}=0for m=1,2,3,4m=1,2,3,4, i.e., ∫GgdG=∫GgyℓdG=0,(ℓ=1,2,3).\mathop{\int }\limits_{{\mathscr{G}}}g{\rm{d}}{\mathscr{G}}=\mathop{\int }\limits_{{\mathscr{G}}}g{y}_{\ell }{\rm{d}}{\mathscr{G}}=0,\hspace{1.0em}\left(\ell =1,2,3).The subspace J˜q(F){\widetilde{J}}_{q}\left({\mathscr{F}})of Jq(F){J}_{q}\left({\mathscr{F}})stands for the set of all f∈Jq(F)f\in {J}_{q}\left({\mathscr{F}})satisfying the orthogonal conditions ∫Ffdy=∫Ff⋅(e3×y)dy=0,∫Ff⋅(eα×y)dy=ω∫Gg˜yαy3dG,(α=1,2),\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f{\rm{d}}y& =& \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{3}\times y){\rm{d}}y=0,\\ \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{\alpha }\times y){\rm{d}}y& =& \omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{g}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},\hspace{1.0em}\left(\alpha =1,2),\end{array}where g˜∈B˜q,q2−1/q(G)\widetilde{g}\in {\widetilde{B}}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}). We then define X˜0≔J˜q(F)∩B˜q,q2−1/q(G).{\widetilde{X}}_{0}:= {\widetilde{J}}_{q}\left({\mathscr{F}})\cap {\widetilde{B}}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}).We set the restriction operator A˜q≔Aq∣X˜0{\widetilde{{\mathcal{A}}}}_{q}:= {{\mathcal{A}}}_{q}{| }_{{\widetilde{X}}_{0}}with its domain given by D(A˜q)≔D(Aq)∩X˜0{\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q}):= {\mathsf{D}}\left({{\mathcal{A}}}_{q})\cap {\widetilde{X}}_{0}. Then the operator −A˜q-{\widetilde{{\mathcal{A}}}}_{q}generates an analytic C0{C}_{0}-semigroup on X˜0{\widetilde{X}}_{0}.Lemma 5.5Let 1<q<∞1\lt q\lt \infty . The induced operator −A˜q≔−Aq∣X˜0-{\widetilde{{\mathcal{A}}}}_{q}:= -{{\mathcal{A}}}_{q}{| }_{{\widetilde{X}}_{0}}with domain D(A˜q)≔D(Aq)∩X˜0{\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q}):= {\mathsf{D}}\left({{\mathcal{A}}}_{q})\cap {\widetilde{X}}_{0}is the generator of an analytic C0{C}_{0}-semigroup in X˜0{\widetilde{X}}_{0}.To prove this lemma, we need the following proposition given in [33, Prop. 2.3].Proposition 5.6Let rrbe a function defined on G{\mathscr{G}}and Gr{{\mathscr{G}}}_{r}be a normal perturbation of G{\mathscr{G}}given byGr≔{s=y+rνG(y):y∈G},{{\mathscr{G}}}_{r}:= \left\{s=y+r{\nu }_{{\mathscr{G}}}(y)\hspace{0.33em}:\hspace{0.33em}y\in {\mathscr{G}}\right\},where ∣r∣L∞(G)| r{| }_{{L}^{\infty }\left({\mathscr{G}})}and ∣∇Gr∣L∞(G)| {\nabla }_{{\mathscr{G}}}r{| }_{{L}^{\infty }\left({\mathscr{G}})}are assumed to be suitably small such that Gr{{\mathscr{G}}}_{r}is contained in the tubular neighborhood of G{\mathscr{G}}. For arbitrary function ζ(y)=a+b×y\zeta (y)=a+b\times y(with constants aaand bb) defined on G{\mathscr{G}}, the equality∫GC^GrνG⋅ζdG=−ω2∫Grζ⋅y′dG\mathop{\int }\limits_{{\mathscr{G}}}{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}r{\nu }_{{\mathscr{G}}}\cdot \zeta {\rm{d}}{\mathscr{G}}=-{\omega }^{2}\mathop{\int }\limits_{{\mathscr{G}}}r\zeta \cdot y^{\prime} {\rm{d}}{\mathscr{G}}holds.ProofThere are only a few words for the proof in [33], but we record the proof here for the reader’s convenience. We denote by Fr{{\mathscr{F}}}_{r}the closed domain surrounded by Gr{{\mathscr{G}}}_{r}. We consider the integral (5.11)I[r]≔∫GrσHGr+ω22∣s′∣2+p0νGr⋅ζdGr,I\left[r]:= \mathop{\int }\limits_{{{\mathscr{G}}}_{r}}\left(\sigma {{\mathscr{H}}}_{{{\mathscr{G}}}_{r}}+\frac{{\omega }^{2}}{2}| s^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{{{\mathscr{G}}}_{r}}\cdot \zeta {\rm{d}}{{\mathscr{G}}}_{r},where s′≔(s1,s2,0)s^{\prime} := \left({s}_{1},{s}_{2},0). Since HGrνGr=ΔGrs{{\mathscr{H}}}_{{{\mathscr{G}}}_{r}}{\nu }_{{{\mathscr{G}}}_{r}}={\Delta }_{{{\mathscr{G}}}_{r}}sfor s∈Grs\in {{\mathscr{G}}}_{r}and ∫Grp0νGr⋅ζdGr=∫Frp0divζds=0{\int }_{{{\mathscr{G}}}_{r}}{p}_{0}{\nu }_{{{\mathscr{G}}}_{r}}\cdot \zeta {\rm{d}}{{\mathscr{G}}}_{r}={\int }_{{{\mathscr{F}}}_{r}}{p}_{0}{\rm{div}}\hspace{0.33em}\zeta {\rm{d}}s=0, integration by parts and the divergence theorem imply (5.12)I[r]=∫Grω22∣s′∣2νGr⋅ζdGr=ω22∫Frdiv(∣s′∣2ζ)dGr=ω2∫Frζ⋅s′ds.I\left[r]=\mathop{\int }\limits_{{{\mathscr{G}}}_{r}}\frac{{\omega }^{2}}{2}| s^{\prime} {| }^{2}{\nu }_{{{\mathscr{G}}}_{r}}\cdot \zeta {\rm{d}}{{\mathscr{G}}}_{r}=\frac{{\omega }^{2}}{2}\mathop{\int }\limits_{{{\mathscr{F}}}_{r}}{\rm{div}}\hspace{0.33em}\left(| s^{\prime} {| }^{2}\zeta ){\rm{d}}{{\mathscr{G}}}_{r}={\omega }^{2}\mathop{\int }\limits_{{{\mathscr{F}}}_{r}}\zeta \cdot s^{\prime} {\rm{d}}s.We next calculate the first variation of I(r)I\left(r). By using the well known formulas δ0HGr=ΔGr+(HG2−2KG)r,δ0∣s′∣2=2νG⋅y′r,{\delta }_{0}{{\mathscr{H}}}_{{{\mathscr{G}}}_{r}}={\Delta }_{{\mathscr{G}}}r+\left({{\mathscr{H}}}_{{\mathscr{G}}}^{2}-2{{\mathscr{K}}}_{{\mathscr{G}}})r,\hspace{1.0em}{\delta }_{0}| s^{\prime} {| }^{2}=2{\nu }_{{\mathscr{G}}}\cdot y^{\prime} r,it follows from (5.11) that δ0I[r]=−∫GℬGrνG⋅ζdG=−∫GC^GrνG⋅ζdG.{\delta }_{0}I\left[r]=-\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}r{\nu }_{{\mathscr{G}}}\cdot \zeta {\rm{d}}{\mathscr{G}}=-\mathop{\int }\limits_{{\mathscr{G}}}{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}r{\nu }_{{\mathscr{G}}}\cdot \zeta {\rm{d}}{\mathscr{G}}.Conversely, the first variation of (5.12) gives δ0I[r]=ω2∫Grζ⋅y′dG{\delta }_{0}I\left[r]={\omega }^{2}\mathop{\int }\limits_{{\mathscr{G}}}r\zeta \cdot y^{\prime} {\rm{d}}{\mathscr{G}}due to a formula δ0∫Frfds=∫GfrdG{\delta }_{0}{\int }_{{{\mathscr{F}}}_{r}}f{\rm{d}}s={\int }_{{\mathscr{G}}}fr{\rm{d}}{\mathscr{G}}. Combining the aforementioned equalities, we obtain the required equality.□We next give the proof of Lemma 5.5.Proof of Lemma 5.5First, we show that the closed subspace X˜0{\widetilde{X}}_{0}of X0{X}_{0}is e−Aqt{e}^{-{{\mathcal{A}}}_{q}t}-invariant, i.e., e−AqtX˜0⊂X˜0{e}^{-{{\mathcal{A}}}_{q}t}{\widetilde{X}}_{0}\subset {\widetilde{X}}_{0}for any t≥0t\ge 0. According to the classical semigroup theory (cf. [20, Thm. 4.5.1]), it suffices to show that there is a real number c∗{c}_{\ast }such that for every λ>c∗\lambda \gt {c}_{\ast }, the space X˜0{\widetilde{X}}_{0}is an invariant subspace of R(λ;−Aq)R\left(\lambda ;\hspace{0.33em}-{{\mathcal{A}}}_{q}), the resolvent of −Aq-{{\mathcal{A}}}_{q}. To this end, we shall consider the following resolvent problem (5.13)λU^∗−Lω,yU^∗+∇Q^∗=F˜,in F,divU^∗=0,in F,PG(2μD(U^∗)νG)=0,on G,2μD(U^∗)νG⋅νG−Q^∗+C^Gh^∗=0,on G,λh^∗−(P0GU^∗)⋅νG=G˜,on G\left\{\begin{array}{ll}\lambda {\widehat{U}}_{\ast }-{L}_{\omega ,y}{\widehat{U}}_{\ast }+\nabla {\widehat{Q}}_{\ast }=\widetilde{F},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{\widehat{U}}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{\widehat{Q}}_{\ast }+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=\widetilde{G},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for given (F˜,G˜)∈X˜0\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}. Here, λ∈Σε,λ∗\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{\ast }}is a resolvent parameter, where ε∈(0,π/2)\varepsilon \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2)and λ∗≥max(λ4,3∣ω∣){\lambda }_{\ast }\ge \max \left({\lambda }_{4},3| \omega | )are constants. From Lemma 5.4, the resolvent set ρ(−Aq)\rho \left(-{{\mathcal{A}}}_{q})of −Aq-{{\mathcal{A}}}_{q}contains Σε,λ∗{\Sigma }_{\varepsilon ,{\lambda }_{\ast }}.Integrating (5.13)5{}_{5}, for m=1,2,3,4m=1,2,3,4, it holds 0=(G˜,φm)G=(λh^∗−(P0GU^∗)⋅νG,φm)G=λ(h^∗,φm)G−∫Fdiv((P0GU^∗)φm)dy.0={\left(\widetilde{G},{\varphi }_{m})}_{{\mathscr{G}}}={\left(\lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}},{\varphi }_{m})}_{{\mathscr{G}}}=\lambda {\left({\widehat{h}}_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}-\mathop{\int }\limits_{{\mathscr{F}}}{\rm{div}}\hspace{0.33em}\left(\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast }){\varphi }_{m}){\rm{d}}y.In the following, we write U^∗≔(U^∗(1),U^∗(2),U^∗(3)){\widehat{U}}_{\ast }:= \left({\widehat{U}}_{\ast }^{\left(1)},{\widehat{U}}_{\ast }^{\left(2)},{\widehat{U}}_{\ast }^{\left(3)}). As div(P0GU^∗)=divU^∗=0{\rm{div}}\hspace{0.33em}\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })={\rm{div}}\hspace{0.33em}{\widehat{U}}_{\ast }=0in F{\mathscr{F}}and ∂ℓφm{\partial }_{\ell }{\varphi }_{m}, ℓ=1,2,3\ell =1,2,3, are constants, we observe ∫Fdiv((P0GU^∗)φm)dy=∫F(div(P0GU^∗))φmdy+∑ℓ=13∫FU^∗(ℓ)−1∣F∣∫FU^∗(ℓ)dy(∂ℓφm)dy=0.\mathop{\int }\limits_{{\mathscr{F}}}{\rm{div}}\hspace{0.33em}\left(\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast }){\varphi }_{m}){\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}\left({\rm{div}}\hspace{0.33em}\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })){\varphi }_{m}{\rm{d}}y+\mathop{\sum }\limits_{\ell =1}^{3}\mathop{\int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(\ell )}-\frac{1}{| {\mathscr{F}}| }\mathop{\int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(\ell )}{\rm{d}}y\right)\left({\partial }_{\ell }{\varphi }_{m}){\rm{d}}y=0.Since λ≠0\lambda \ne 0, this gives (h^∗,φm)G=0{\left({\widehat{h}}_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}=0for m=1,2,3,4m=1,2,3,4.Employing the similar argument as mentioned earlier, we can show U^∗{\widehat{U}}_{\ast }satisfies the orthogonal conditions. In fact, integrating (5.13)1{}_{1}, for ℓ=1,2,3\ell =1,2,3, we have 0=(F˜,eℓ)F=(λU^∗−μΔU^∗+2ω(e3×U^∗)+∇Q^∗,eℓ)F=λ(U^∗,eℓ)F+2ω(e3×U^∗,eℓ)F+(C^Gh^∗,νG⋅eℓ)G.0={\left(\widetilde{F},{e}_{\ell })}_{{\mathscr{F}}}={\left(\lambda {\widehat{U}}_{\ast }-\mu \Delta {\widehat{U}}_{\ast }+2\omega \left({e}_{3}\times {\widehat{U}}_{\ast })+\nabla {\widehat{Q}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}=\lambda {\left({\widehat{U}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}+2\omega {\left({e}_{3}\times {\widehat{U}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}+{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot {e}_{\ell })}_{{\mathscr{G}}}.By Proposition 5.6, it holds (C^Gh^∗,νG⋅eα)G=−ω2∫Gh^∗yαdG,(α=1,2),(C^Gh^∗,νG⋅e3)G=0,{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot {e}_{\alpha })}_{{\mathscr{G}}}=-{\omega }^{2}\mathop{\int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{\alpha }{\rm{d}}{\mathscr{G}},\hspace{1.0em}\left(\alpha =1,2),\hspace{1.0em}{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot {e}_{3})}_{{\mathscr{G}}}=0,and thus, we obtain λ∫FU^∗(1)dy−2ω∫FU^∗(2)dy−ω2∫Gh^∗y1dG=0,λ∫FU^∗(2)dy+2ω∫FU^∗(1)dy−ω2∫Gh^∗y2dG=0,λ∫FU^∗(3)dy=0.\begin{array}{rcl}\lambda \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(1)}{\rm{d}}y-2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(2)}{\rm{d}}y-{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{\rm{d}}{\mathscr{G}}& =& 0,\\ \lambda \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(2)}{\rm{d}}y+2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(1)}{\rm{d}}y-{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{\rm{d}}{\mathscr{G}}& =& 0,\\ \lambda \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(3)}{\rm{d}}y& =& 0.\end{array}Recalling that h^∗{\widehat{h}}_{\ast }satisfies (h^∗,φm)G=0{\left({\widehat{h}}_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}=0for m=1,2,3,4m=1,2,3,4, we obtain (U^∗,eℓ)F=0{\left({\widehat{U}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}=0, ℓ=1,2,3\ell =1,2,3, due to λ≠0,±2iω\lambda \ne 0,\pm 2i\omega .We next multiply the equation (5.13)1{}_{1}by eℓ×y{e}_{\ell }\times y, ℓ=1,2,3\ell =1,2,3, and integrate over F{\mathscr{F}}, which gives (F˜,eℓ×y)F=λ(U^∗,eℓ×y)F−2ωδℓ,3∫FU^∗⋅y′dy+2ω(e3×U^∗,eℓ×y)F+(C^Gh^∗,νG⋅(eℓ×y))G.{\left(\widetilde{F},{e}_{\ell }\times y)}_{{\mathscr{F}}}=\lambda {\left({\widehat{U}}_{\ast },{e}_{\ell }\times y)}_{{\mathscr{F}}}-2\omega {\delta }_{\ell ,3}\left(\mathop{\int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }\cdot y^{\prime} {\rm{d}}y\right)+2\omega {\left({e}_{3}\times {\widehat{U}}_{\ast },{e}_{\ell }\times y)}_{{\mathscr{F}}}+{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{\ell }\times y))}_{{\mathscr{G}}}.Here, δℓ,3{\delta }_{\ell ,3}stands for the Kronecker delta. Since Proposition 5.6 gives (C^Gh^∗,νG⋅(e1×y))G=ω2∫Gh^∗y2y3dG,(C^Gh^∗,νG⋅(e2×y))G=−ω2∫Gh^∗y1y3dG,(C^Gh^∗,νG⋅(e3×y))G=0,\begin{array}{rcl}{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{1}\times y))}_{{\mathscr{G}}}& =& {\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{2}\times y))}_{{\mathscr{G}}}& =& -{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{3}\times y))}_{{\mathscr{G}}}& =& 0,\end{array}it holds (F˜,e1×y)F=λ(U^∗,e1×y)F−2ω∫FU^∗(1)y3dy+ω2∫Gh^∗y2y3dG,(F˜,e2×y)F=λ(U^∗,e2×y)F−2ω∫FU^∗(2)y3dy−ω2∫Gh^∗y1y3dG,(F˜,e3×y)F=λ(U^∗,e3×y)F.\begin{array}{rcl}{\left(\widetilde{F},{e}_{1}\times y)}_{{\mathscr{F}}}& =& \lambda {\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(1)}{y}_{3}{\rm{d}}y+{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left(\widetilde{F},{e}_{2}\times y)}_{{\mathscr{F}}}& =& \lambda {\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(2)}{y}_{3}{\rm{d}}y-{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left(\widetilde{F},{e}_{3}\times y)}_{{\mathscr{F}}}& =& \lambda {\left({\widehat{U}}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}.\end{array}Conversely, we have ω∫GG˜yαy3dG=ω∫G(λh^∗−(P0GU^∗)⋅νG)yαy3dG=λω∫Gh^∗yαy3dG−ω∫F(U^∗(α)y3+U^∗(3)yα)dy\omega \mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=\omega \mathop{\int }\limits_{{\mathscr{G}}}(\lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=\lambda \omega \mathop{\int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}-\omega \mathop{\int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(\alpha )}{y}_{3}+{\widehat{U}}_{\ast }^{\left(3)}{y}_{\alpha }){\rm{d}}ywith α=1,2\alpha =1,2. Since F˜\widetilde{F}and G˜\widetilde{G}satisfy the conditions: ∫FF˜⋅(eα×y)dy=ω∫GG˜yαy3dG,(α=1,2),∫FF˜⋅(e3×y)dy=0,\mathop{\int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{\alpha }\times y){\rm{d}}y=\omega \mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},\hspace{1.0em}\left(\alpha =1,2),\hspace{1.0em}\mathop{\int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{3}\times y){\rm{d}}y=0,we observe (5.14)λ(U^∗,e1×y)F−ω∫Gh^∗y1y3dG−ω∫F(U^∗(1)y3−U^∗(3)y1)dy−ω∫Gh^∗y2y3dG=0,λ(U^∗,e2×y)F−ω∫Gh^∗y2y3dG+ω∫F(U^∗(3)y2−U^∗(2)y3)dy−ω∫Gh^∗y1y3dG=0,λ(U^∗,e3×y)F=0.\begin{array}{l}\lambda \left({\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)-\omega \left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(1)}{y}_{3}-{\widehat{U}}_{\ast }^{\left(3)}{y}_{1}){\rm{d}}y-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)=0,\\ \lambda \left({\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)+\omega \left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(3)}{y}_{2}-{\widehat{U}}_{\ast }^{\left(2)}{y}_{3}){\rm{d}}y-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)=0,\\ \lambda {\left({\widehat{U}}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}=0.\end{array}Noting that U^∗(1)y3−U^∗(3)y1=U^∗⋅(e2×y),U^∗(2)y3−U^∗(3)y2=U^∗⋅(e1×y),{\widehat{U}}_{\ast }^{\left(1)}{y}_{3}-{\widehat{U}}_{\ast }^{\left(3)}{y}_{1}={\widehat{U}}_{\ast }\cdot \left({e}_{2}\times y),\hspace{1.0em}{\widehat{U}}_{\ast }^{\left(2)}{y}_{3}-{\widehat{U}}_{\ast }^{\left(3)}{y}_{2}={\widehat{U}}_{\ast }\cdot \left({e}_{1}\times y),it follows from (5.14)1,2{\left(5.14)}_{1,2}that λ(U^∗,e1×y)F−ω∫Gh^∗y1y3dG−ω(U^∗,e2×y)F−ω∫Gh^∗y2y3dG=0,λ(U^∗,e2×y)F−ω∫Gh^∗y2y3dG+ω(U^∗,e1×y)F−ω∫Gh^∗y1y3dG=0.\begin{array}{rcl}\lambda \left({\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)-\omega \left({\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)& =& 0,\\ \lambda \left({\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)+\omega \left({\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)& =& 0.\end{array}Hence, by λ≠0,±iω\lambda \ne 0,\pm i\omega , we arrive at (U^∗,eα×y)F−ω∫Gh^∗yαy3dG=0,(α=1,2),(U^∗,e3×y)F=0,{\left({\widehat{U}}_{\ast },{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=0,\hspace{1.0em}\left(\alpha =1,2),\hspace{1.0em}{\left({\widehat{U}}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}=0,which implies (U^∗,h^∗)∈D(A˜q)\left({\widehat{U}}_{\ast },{\widehat{h}}_{\ast })\in {\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q}). Accordingly, we have the R(λ;−Aq)R\left(\lambda ;-{{\mathcal{A}}}_{q})-invariance of X˜0{\widetilde{X}}_{0}for any λ∈Σε,λ∗\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{\ast }}, which implies Σε,λ∗⊂ρ(−A˜q){\Sigma }_{\varepsilon ,{\lambda }_{\ast }}\subset \rho \left(-{\widetilde{{\mathcal{A}}}}_{q})(cf. [10, Prop. A.2.8]). Hence, the C0{C}_{0}-semigroup {e−Aqt∣X˜0}t≥0{\left\{{e}^{-{{\mathcal{A}}}_{q}t}{| }_{{\widetilde{X}}_{0}}\right\}}_{t\ge 0}is indeed analytic.□In the following, we denote the induced C0{C}_{0}-semigroup by {e−A˜qt}t≥0≔{e−Aqt∣X˜0}t≥0.\{{e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}{\}}_{t\ge 0}:= \{{e}^{-{{\mathcal{A}}}_{q}t}{| }_{{\widetilde{X}}_{0}}{\}}_{t\ge 0}.As F{\mathscr{F}}and G{\mathscr{G}}are compact, we can prove that 0∈ρ(A˜q)0\in \rho \left({\widetilde{{\mathcal{A}}}}_{q}), which immediately implies the exponential stability of the analytic C0{C}_{0}-semigroup generated by −A˜q-{\widetilde{{\mathcal{A}}}}_{q}.Theorem 5.7Let 1<q<∞1\lt q\lt \infty . For any (F˜,G˜)∈X˜0\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}and t>0t\gt 0, there exist positive constants C and β∗{\beta }_{\ast }such that∣e−A˜qt(F˜,G˜)∣X0≤Ce−β∗t∣(F˜,G˜)∣X0| {e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}\left(\widetilde{F},\widetilde{G}){| }_{{X}_{0}}\le C{e}^{-{\beta }_{\ast }t}| \left(\widetilde{F},\widetilde{G}){| }_{{X}_{0}}is valid, i.e., {e−A˜qt}t≥0{\left\{{e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}\right\}}_{t\ge 0}is exponentially stable on X˜0{\widetilde{X}}_{0}.ProofFrom Lemma 5.5, there exists λ∗>0{\lambda }_{\ast }\gt 0such that Σε,λ∗⊂ρ(−A˜q){\Sigma }_{\varepsilon ,{\lambda }_{\ast }}\subset \rho \left(-{\widetilde{{\mathcal{A}}}}_{q})for ε∈(0,π/2)\varepsilon \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2). It remains to prove out result for λ∈Qλ∗≔{λ∈C:Reλ≥0,∣λ∣≤λ∗}\lambda \in {Q}_{{\lambda }_{\ast }}:= \left\{\lambda \in {\mathbb{C}}\hspace{0.33em}:\hspace{0.33em}{\rm{Re}}\hspace{0.33em}\lambda \ge 0,| \lambda | \le {\lambda }_{\ast }\right\}. We define R∗≔R(2λ∗;−A˜q):X˜0→X1∩X˜0⊂X˜0.{R}_{\ast }:= R\left(2{\lambda }_{\ast };-{\widetilde{{\mathcal{A}}}}_{q})\hspace{0.33em}:\hspace{0.33em}{\widetilde{X}}_{0}\to {X}_{1}\cap {\widetilde{X}}_{0}\subset {\widetilde{X}}_{0}.Since F{\mathscr{F}}and G{\mathscr{G}}are compact, it follows from the Rellich theorem that R∗{R}_{\ast }is a compact operator from X˜0{\widetilde{X}}_{0}into itself. For any λ∈Qλ∗\lambda \in {Q}_{{\lambda }_{\ast }}, rewriting I+A˜qI+{\widetilde{{\mathcal{A}}}}_{q}by (λI+A˜q)(F˜,G˜)=(I+(λ−2λ∗)R∗)(2λ∗I+Aq)(F˜,G˜)\left(\lambda I+{\widetilde{{\mathcal{A}}}}_{q})\left(\widetilde{F},\widetilde{G})=\left(I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast })\left(2{\lambda }_{\ast }I+{{\mathcal{A}}}_{q})\left(\widetilde{F},\widetilde{G})for (F˜,G˜)∈X˜0\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}, we observe that Qλ∗⊂ρ(−A˜q){Q}_{{\lambda }_{\ast }}\subset \rho \left(-{\widetilde{{\mathcal{A}}}}_{q})follows from the Fredholm alternative theorem and the injection of I+(λ−2λ∗)R∗I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast }. To see this, for any λ∈Qλ∗\lambda \in {Q}_{{\lambda }_{\ast }}, take (F˜,G˜)∈Ker(I+(λ−2λ∗)R∗)⊂X˜0\left(\widetilde{F},\widetilde{G})\in {\rm{Ker}}\hspace{0.33em}\left(I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast })\subset {\widetilde{X}}_{0}, i.e., (I+(λ−2λ∗)R∗)(F˜,G˜)=0for any(F˜,G˜)∈X˜0.\left(I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast })\left(\widetilde{F},\widetilde{G})=0\hspace{1.0em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}.By the definition of R∗{R}_{\ast }, we see that (F˜,G˜)\left(\widetilde{F},\widetilde{G})belongs to D(A˜q){\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q})and satisfies (5.15)(λ+A˜q)(F˜,G˜)=0for anyλ∈Qλ∗.\left(\lambda +{\widetilde{{\mathcal{A}}}}_{q})\left(\widetilde{F},\widetilde{G})=0\hspace{1.0em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}\lambda \in {Q}_{{\lambda }_{\ast }}.Notice that the equation (5.15) is equivalent to (5.16)λF˜−Lω,yF˜+∇K(F˜,G˜)=0,in F,PG(2μD(F˜)νG)=0,on G,2μD(F˜)νG⋅νG−K(F˜,G˜)+C^GG˜=0,on G,λG˜−(P0GF˜)⋅νG=0,on G\left\{\begin{array}{ll}\lambda \widetilde{F}-{L}_{\omega ,y}\widetilde{F}+\nabla K\left(\widetilde{F},\widetilde{G})=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widetilde{F}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widetilde{F}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-K\left(\widetilde{F},\widetilde{G})+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widetilde{G}-\left({P}_{0}^{{\mathscr{G}}}\widetilde{F})\cdot {\nu }_{{\mathscr{G}}}=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.with the orthogonal conditions (5.17)∫FF˜dy=∫FF˜⋅(e3×y)dy=0,∫FF˜⋅(eα×y)dy=ω∫GG˜yαy3dG,(α=1,2),∫GG˜dG=∫GG˜yℓdG=0,(ℓ=1,2,3).\left\{\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\widetilde{F}{\rm{d}}y& =& \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{3}\times y){\rm{d}}y=0,\\ \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{\alpha }\times y){\rm{d}}y& =& \omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},& \left(\alpha =1,2),\\ \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}{\rm{d}}{\mathscr{G}}& =& \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\ell }{\rm{d}}{\mathscr{G}}=0,& \left(\ell =1,2,3).\end{array}\right.In the following, we may assume (F˜,G˜)∈D(A˜2)\left(\widetilde{F},\widetilde{G})\in {\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{2}). In fact, the boundedness of F{\mathscr{F}}implies D(A˜q)↪D(A˜2){\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{2})for 2≤q<∞2\le q\lt \infty . Besides, when 1<q≤21\lt q\le 2, by the bootstrap argument and Sobolev embedding theorem, we see (F˜,G˜)∈D(A˜2)\left(\widetilde{F},\widetilde{G})\in {\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{2})as well. Using the divergence theorem and (5.17), it follows from (5.16) that (5.18)0=λ∣F˜∣L2(F)2+2μ∣D(F˜)∣L2(F)2+(C^GG˜,F˜⋅νG)G=λ∣F˜∣L2(F)2+2μ∣D(F˜)∣L2(F)2+λ¯(C^GG˜,G˜)G=λ∣F˜∣L2(F)2+2μ∣D(F˜)∣L2(F)2+λ¯ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2,\begin{array}{rcl}0& =& \lambda | \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+2\mu | D\left(\widetilde{F}){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G},\widetilde{F}\cdot {\nu }_{{\mathscr{G}}})}_{{\mathscr{G}}}\\ & =& \lambda | \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+2\mu | D\left(\widetilde{F}){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\overline{\lambda }{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G},\widetilde{G})}_{{\mathscr{G}}}\\ & =& \lambda | \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+2\mu | D\left(\widetilde{F}){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\overline{\lambda }\left\{{\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\right\},\end{array}where ΨG{\Psi }_{{\mathscr{G}}}is the quadratic form given in (1.12). Now, we decompose F˜=F˜⊥+∑α=1,2dα[G˜](eα×y)\widetilde{F}={\widetilde{F}}^{\perp }+\sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]\left({e}_{\alpha }\times y)with F˜⊥{\widetilde{F}}^{\perp }satisfying ∫FF˜⊥dy=∫FF˜⊥⋅(eℓ×y)dy=0,(ℓ=1,2,3),\mathop{\int }\limits_{{\mathscr{F}}}{\widetilde{F}}^{\perp }{\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}{\widetilde{F}}^{\perp }\cdot \left({e}_{\ell }\times y){\rm{d}}y=0,\hspace{1.0em}\left(\ell =1,2,3),and ∣F˜∣L2(F)2=∣F˜⊥∣L2(F)2+∑α=1,2dα[G˜]2Sα.| \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}=| {\widetilde{F}}^{\perp }{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\sum _{\alpha =1,2}{d}_{\alpha }{\left[\widetilde{G}]}^{2}{{\mathcal{S}}}_{\alpha }.Then, (5.18) becomes 0=λ∣F˜⊥∣L2(F)2+∑α=1,2dα[G˜]2Sα+2μ∣D(F˜⊥)∣L2(F)2+λ¯ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2.0=\lambda \left(| {\widetilde{F}}^{\perp }{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\sum _{\alpha =1,2}{d}_{\alpha }{\left[\widetilde{G}]}^{2}{{\mathcal{S}}}_{\alpha }\right)+2\mu | D\left({\widetilde{F}}^{\perp }){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\overline{\lambda }\left\{{\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\right\}.Taking the real part yields 0=2μ∣D(F˜⊥)∣L2(F)2+(Reλ)∣F˜⊥∣L2(F)2+∑α=1,2dα[G˜]2Sα+ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2.0=2\mu | D\left({\widetilde{F}}^{\perp }){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\left({\rm{Re}}\hspace{0.33em}\lambda )\left\{| {\widetilde{F}}^{\perp }{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\sum _{\alpha =1,2}{d}_{\alpha }{\left[\widetilde{G}]}^{2}{{\mathcal{S}}}_{\alpha }+{\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\right\}.According to Assumption 1.1, we see that Reλ≥0{\rm{Re}}\hspace{0.33em}\lambda \ge 0implies D(F˜⊥)=0D\left({\widetilde{F}}^{\perp })=0in F{\mathscr{F}}. Hence, it follows from the second Korn inequality that F˜⊥=0{\widetilde{F}}^{\perp }=0in F{\mathscr{F}}. Then, the first equation of (5.16) takes the form λ∑α=1,2dα[G˜](eα×y)−Lω,y∑α=1,2dα[G˜](eα×y)+∇K(F˜,G˜)=0in F.\lambda \sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]\left({e}_{\alpha }\times y)-{L}_{\omega ,y}\left(\sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]\left({e}_{\alpha }\times y)\right)+\nabla K\left(\widetilde{F},\widetilde{G})=0\hspace{1.0em}\hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{}.Taking the curl{\rm{curl}}in this equation leads to λ∑α=1,2dα[G˜]eα−ω(d2[G˜]e1−d1[G˜]e2)=0.\lambda \sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]{e}_{\alpha }-\omega ({d}_{2}\left[\widetilde{G}]{e}_{1}-{d}_{1}\left[\widetilde{G}]{e}_{2})=0.Hence, we obtain λd1[G˜]=ωd2[G˜]\lambda {d}_{1}\left[\widetilde{G}]=\omega {d}_{2}\left[\widetilde{G}]and λd2[G˜]=−ωd1[G˜]\lambda {d}_{2}\left[\widetilde{G}]=-\omega {d}_{1}\left[\widetilde{G}]. Besides, it follows from (5.17)2{}_{2}that λdα[G˜]=−λωSα∫GG˜yαy3dG=−λSα∫F∑β=1,2dβ[G˜](eβ×y)⋅(eα×y)dy,(α=1,2).\lambda {d}_{\alpha }\left[\widetilde{G}]=-\lambda \frac{\omega }{{{\mathcal{S}}}_{\alpha }}\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=-\frac{\lambda }{{{\mathcal{S}}}_{\alpha }}\mathop{\int }\limits_{{\mathscr{F}}}\sum _{\beta =1,2}{d}_{\beta }\left[\widetilde{G}]\left({e}_{\beta }\times y)\cdot \left({e}_{\alpha }\times y){\rm{d}}y,\hspace{1.0em}\left(\alpha =1,2).Setting S˜α=∫F(yα2−y32)dy,(α=1,2),{\widetilde{{\mathcal{S}}}}_{\alpha }=\mathop{\int }\limits_{{\mathscr{F}}}({y}_{\alpha }^{2}-{y}_{3}^{2}){\rm{d}}y,\hspace{1.0em}\left(\alpha =1,2),we easily observe λd1[G˜]=−ωS˜1S1d2[G˜]=−λS˜1S1d1[G˜],λd2[G˜]=ωS˜2S2d1[G˜]=−λS˜2S2d2[G˜],\lambda {d}_{1}\left[\widetilde{G}]=-\frac{\omega {\widetilde{{\mathcal{S}}}}_{1}}{{{\mathcal{S}}}_{1}}{d}_{2}\left[\widetilde{G}]=-\lambda \frac{{\widetilde{{\mathcal{S}}}}_{1}}{{{\mathcal{S}}}_{1}}{d}_{1}\left[\widetilde{G}],\hspace{1.0em}\lambda {d}_{2}\left[\widetilde{G}]=\frac{\omega {\widetilde{{\mathcal{S}}}}_{2}}{{{\mathcal{S}}}_{2}}{d}_{1}\left[\widetilde{G}]=-\lambda \frac{{\widetilde{{\mathcal{S}}}}_{2}}{{{\mathcal{S}}}_{2}}{d}_{2}\left[\widetilde{G}],i.e., d1[G˜]=d2[G˜]=0{d}_{1}\left[\widetilde{G}]={d}_{2}\left[\widetilde{G}]=0, as follows: S˜α+Sα=∫F((yα2−y32)+(∣y∣2−yα2))dy=∫F∣y′∣2dy≠0.{\widetilde{{\mathcal{S}}}}_{\alpha }+{{\mathcal{S}}}_{\alpha }=\mathop{\int }\limits_{{\mathscr{F}}}(({y}_{\alpha }^{2}-{y}_{3}^{2})+\left(| y{| }^{2}-{y}_{\alpha }^{2})){\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}| y^{\prime} {| }^{2}{\rm{d}}y\ne 0.Thus, we arrive at F˜=0\widetilde{F}=0in F{\mathscr{F}}, which, combined with (5.16)1{}_{1}, implies that the pressure term K(F˜,G˜)K\left(\widetilde{F},\widetilde{G})is equal to some constant p0{{\mathsf{p}}}_{0}. From (5.16)2{}_{2}, we have C^GG˜=p0{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}={{\mathsf{p}}}_{0}on G{\mathscr{G}}, but it follows from ∫GC^GG˜dG=0{\int }_{{\mathscr{G}}}{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}{\rm{d}}{\mathscr{G}}=0that p0=0{{\mathsf{p}}}_{0}=0, i.e., C^GG˜=0{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}=0on G{\mathscr{G}}. Hence, we observe that 0=(C^GG˜,G˜)G=ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2≥c∣G˜∣L2(G)2+ω2S3∫GG˜∣y′∣2dG2,0={\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G},\widetilde{G})}_{{\mathscr{G}}}={\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\ge c| \widetilde{G}{| }_{{L}^{2}\left({\mathscr{G}})}^{2}+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2},which implies G˜=0\widetilde{G}=0on G{\mathscr{G}}. Therefore, there are no eigenvalues λ∈Qλ∗\lambda \in {Q}_{{\lambda }_{\ast }}of −A˜2-{\widetilde{{\mathcal{A}}}}_{2}. This completes the proof.□By using Theorem 5.7, we complete the decay estimate of solutions to (5.3).Theorem 5.8Assume that 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Let v∗{v}_{\ast }and η∗{\eta }_{\ast }be functions obtained in Theorem 5.2. Then, the solution (u∗,h∗)\left({u}_{\ast },{h}_{\ast })to (5.3) enjoys the estimate∣eε0t(u∗,h∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({u}_{\ast },{h}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).with a constant C independent of T.ProofBy the variation of constants formula, we see that (U∗,h∗)(⋅,t)=∫0te−A˜q(t−s)(2λ2f˜w(⋅,s),2λ2η˜∗(⋅,s))ds\left({U}_{\ast },{h}_{\ast })\left(\cdot ,t)=\underset{0}{\overset{t}{\int }}{e}^{-{\widetilde{{\mathcal{A}}}}_{q}\left(t-s)}(2{\lambda }_{2}{\widetilde{f}}_{w}\left(\cdot ,s),2{\lambda }_{2}{\widetilde{\eta }}_{\ast }\left(\cdot ,s)){\rm{d}}sare solutions to (5.7) with Q∗=K(U∗,h∗){Q}_{\ast }=K\left({U}_{\ast },{h}_{\ast }). Then, Theorem 5.7 yields ∣(U∗,h∗)(⋅,t)∣X0≤Cλ2∫0te−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0ds≤Cλ2∫0te−β∗(t−s)ds1/p′∫0te−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0pds1/p≤Cλ2β∗−1/p′∫0te−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0pds1/p.\begin{array}{rcl}| \left({U}_{\ast },{h}_{\ast })\left(\cdot ,t){| }_{{X}_{0}}& \le & C{\lambda }_{2}\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}{\rm{d}}s\\ & \le & C{\lambda }_{2}{\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}{\rm{d}}s\right)}^{1\text{/}p^{\prime} }{\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & \le & C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }{\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}^{p}{\rm{d}}s\right)}^{1\text{/}p}.\end{array}Hence, for every 0<ε0<β∗/p0\lt {\varepsilon }_{0}\lt {\beta }_{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}p, it holds (5.19)∫0T(eε0t∣(U∗,h∗)(⋅,t)∣X0)pdt≤Cλ2β∗−1/p′∫0T∫0teε0tpe−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0pdsdt=Cλ2β∗−1/p′∫0T∫0te−(β∗−ε0p)(t−s)(eε0s∣(f˜w,η˜∗)(⋅,s)∣X0)pdsdt=Cλ2β∗−1/p′∫0T(eε0s∣(f˜w,η˜∗)(⋅,s)∣X0)p∫sTe−(β∗−ε0p)(t−s)dtds=Cλ2β∗−1/p′(β∗−ε0p)−1∫0T(eε0s∣(f˜w,η˜∗)(⋅,s)∣X0)pds.\begin{array}{rcl}\underset{0}{\overset{T}{\displaystyle \int }}{({e}^{{\varepsilon }_{0}t}| \left({U}_{\ast },{h}_{\ast })\left(\cdot ,t){| }_{{X}_{0}})}^{p}{\rm{d}}t& \le & C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }\underset{0}{\overset{T}{\displaystyle \int }}\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{{\varepsilon }_{0}tp}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}^{p}{\rm{d}}s\right){\rm{d}}t\\ & =& C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }\underset{0}{\overset{T}{\displaystyle \int }}\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-\left({\beta }_{\ast }-{\varepsilon }_{0}p)\left(t-s)}{({e}^{{\varepsilon }_{0}s}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}})}^{p}{\rm{d}}s\right){\rm{d}}t\\ & =& C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }\underset{0}{\overset{T}{\displaystyle \int }}{({e}^{{\varepsilon }_{0}s}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}})}^{p}\left(\underset{s}{\overset{T}{\displaystyle \int }}{e}^{-\left({\beta }_{\ast }-{\varepsilon }_{0}p)\left(t-s)}{\rm{d}}t\right){\rm{d}}s\\ & =& C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }{\left({\beta }_{\ast }-{\varepsilon }_{0}p)}^{-1}\underset{0}{\overset{T}{\displaystyle \int }}{({e}^{{\varepsilon }_{0}s}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}})}^{p}{\rm{d}}s.\end{array}Besides, we have ∣(f˜w,η˜∗)(⋅,s)∣X0≤∣f˜w∣Lq(F)+∣η˜∗∣Bq,q2−1/q(G)≤C(∣v˜∗∣Lq(F)+(1+ω)∣η˜∗∣Bq,q2−1/q(G))≤C(∣v∗∣Lq(F)+(1+ω)∣η∗∣Bq,q2−1/q(G))\begin{array}{rcl}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}& \le & | {\widetilde{f}}_{w}{| }_{{L}^{q}\left({\mathscr{F}})}+| {\widetilde{\eta }}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}\\ & \le & C(| {\widetilde{v}}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+\left(1+\omega )| {\widetilde{\eta }}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})})\\ & \le & C(| {v}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+\left(1+\omega )| {\eta }_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})})\end{array}for any s∈(0,T)s\in \left(0,T). Hence, it follows from the estimate (5.19) and Corollary 5.3 that (5.20)∣eε0t(U∗,h∗)(⋅,t)∣Lp(J;X0)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F))| {e}^{{\varepsilon }_{0}t}\left({U}_{\ast },{h}_{\ast })\left(\cdot ,t){| }_{{L}^{p}\left(J;{X}_{0})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})for t∈(0,T)t\in \left(0,T)and ε0∈(0,β∗/p){\varepsilon }_{0}\in \left(0,{\beta }_{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}p), where CCis independent of TTand tt. Besides, by (5.7)5{\left(5.7)}_{5}, we also obtain the estimate (5.21)∣eε0th∗(⋅,t)∣Fp,q,δ1−1/(2q)(J;Lq(G))≤C∣eε0th∗(⋅,t)∣Hδ1,p(J;Lq(F))∩Lδp(J;H2,q(F))≤C(∣eε0t∂sh∗(⋅,t)∣Lδp(J;Lq(F))+∣eε0th∗(⋅,t)∣Lδp(J;H2,q(F)))≤C(∣eε0t(U∗,η˜∗)(⋅,t)∣Lδp(J;Lq(F))+∣eε0th∗(⋅,t)∣Lδp(J;H2,q(F)))≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F))\begin{array}{rcl}| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{F}_{p,q,\delta }^{1-1\text{/}\left(2q)}\left(J;{L}^{q}\left({\mathscr{G}}))}& \le & C| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{H}_{\delta }^{1,p}\left(J;{L}^{q}\left({\mathscr{F}}))\cap {L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}}))}\\ & \le & C(| {e}^{{\varepsilon }_{0}t}{\partial }_{s}{h}_{\ast }\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}}))})\\ & \le & C(| {e}^{{\varepsilon }_{0}t}\left({U}_{\ast },{\widetilde{\eta }}_{\ast })\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}}))})\\ & \le & C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})\end{array}with a constant CCindependent of ttand TT. If (U˜∗,h˜∗)\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast })satisfies the shifted equations ∂tU˜∗+2λ2U˜∗−Lω,yU˜∗+∇K(U˜∗,h˜∗)=2λ2(f˜w+U∗),in F,divU˜∗=0,in F,PG(2μD(U˜∗)νG)=0,on G,2μD(U˜∗)νG⋅νG−K(U˜∗,h˜∗)+C^Gh˜∗=0,on G,∂th˜∗+2λ2h˜∗−(P0GU˜∗)⋅νG=2λ2(η˜∗+h∗),on G,U˜∗(0)=0,in F,h˜∗(0)=0,on G,\left\{\begin{array}{ll}{\partial }_{t}{\widetilde{U}}_{\ast }+2{\lambda }_{2}{\widetilde{U}}_{\ast }-{L}_{\omega ,y}{\widetilde{U}}_{\ast }+\nabla K\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast })=2{\lambda }_{2}({\widetilde{f}}_{w}+{U}_{\ast }),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{\widetilde{U}}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({\widetilde{U}}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({\widetilde{U}}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-K\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast })+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widetilde{h}}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{\widetilde{h}}_{\ast }+2{\lambda }_{2}{\widetilde{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widetilde{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=2{\lambda }_{2}\left({\widetilde{\eta }}_{\ast }+{h}_{\ast }),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\widetilde{U}}_{\ast }\left(0)=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\widetilde{h}}_{\ast }\left(0)=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.we obtain from Corollary 5.3, (5.20), and (5.21) that ∣eε0t(U˜∗,h˜∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).Noting that U∗{U}_{\ast }and h∗{h}_{\ast }satisfy (5.7), the uniqueness of solutions implies U˜∗=U∗{\widetilde{U}}_{\ast }={U}_{\ast }and h˜∗=h∗{\widetilde{h}}_{\ast }={h}_{\ast }for any t∈(0,T)t\in \left(0,T), and hence, we observe ∣eε0t(U∗,h∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({U}_{\ast },{h}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).Recalling the decomposition (5.6), we arrive at ∣eε0t(u∗,h∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({u}_{\ast },{h}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).This completes the proof of Theorem 5.8.□Step 3: The completion of the proof of Theorem 5.1. Finally, let us derive the estimates of (u,q,h)\left(u,q,h). Recalling (5.4) and (5.5), it follows from Theorems 5.2 and 5.8 that ∣eε0t∂t(u,h)∣Lδp(J;X0)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}{\partial }_{t}\left(u,h){| }_{{L}_{\delta }^{p}\left(J;{X}_{0})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).Besides, we also have ∣eε0t(u,h)∣Lδp(J;X1)≤∣eε0t(u,h)∣Lδp(J;Lq(F)×Lq(G))+∑j=1,2∣eε0t∇ju∣Lδp(J;Lq(F))+∑k=1,2,3∣eε0t∇kh∣Lδp(J;Lq(G))≤C[∣eε0t(v∗,η∗)∣Lδp(J;H2,q(F)×Bq,q3−1/q(G))+∣eε0t(u∗,h∗)∣Lδp(J;H2,q(F)×Bq,q3−1/q(G))+∫0T(eε0s∣(u∗(⋅,s),1)F∣)pds1/p+∑α=1,2∫0T(eε0s∣(u∗(⋅,s),eα×y)F−ω∫Gh∗(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u∗(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h∗(⋅,s),φm)G∣)pds1/p≤C[∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)+∫0T(eε0s∣(u∗(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u∗(⋅,s),eα×y)F−ω∫Gh∗(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u∗(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h∗(⋅,s),φm)G∣)pds1/p,\begin{array}{rcl}| {e}^{{\varepsilon }_{0}t}\left(u,h){| }_{{L}_{\delta }^{p}\left(J;{X}_{1})}& \le & | {e}^{{\varepsilon }_{0}t}\left(u,h){| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}})\times {L}^{q}\left({\mathscr{G}}))}+\displaystyle \sum _{j=1,2}| {e}^{{\varepsilon }_{0}t}{\nabla }^{j}u{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+\displaystyle \sum _{k=1,2,3}| {e}^{{\varepsilon }_{0}t}{\nabla }^{k}h{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{G}}))}\\ & \le & C{[}| {e}^{{\varepsilon }_{0}t}\left({v}_{\ast },{\eta }_{\ast }){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}}))}+| {e}^{{\varepsilon }_{0}t}\left({u}_{\ast },{h}_{\ast }){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}}))}\\ & & +{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}\left(\underset{0}{\overset{T}{\displaystyle \int }}({e}^{{\varepsilon }_{0}s}\phantom{\rule[-7pt]{}{0ex}}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}\right.\\ & & {\left.{\left.\left.-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & & \left.+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({h}_{\ast }\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right]\le C{[}| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}\\ & & +| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & & +\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left({u}_{\ast }\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & & \left.+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({h}_{\ast }\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right],\end{array}where CCis independent of TT. Finally, we estimate eε0th{e}^{{\varepsilon }_{0}t}hin the Fp,q,δ2−/q(J;Lq(G)){F}_{p,q,\delta }^{2-\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))-norm. It holds ∣eε0th∣Fp,q,δ2−1/q(J;Lq(G))≤C∣eε0th∣Hδ2,p(J;Lq(F))∩Hδ1,p(J;H2,q(F))≤C(∣eε0t∂t2h∣Lδp(J;Lq(F))+∣eε0th∣Hδ1,p(J;H2,q(F))).| {e}^{{\varepsilon }_{0}t}h{| }_{{F}_{p,q,\delta }^{2-1\text{/}q}\left(J;{L}^{q}\left({\mathscr{G}}))}\le C| {e}^{{\varepsilon }_{0}t}h{| }_{{H}_{\delta }^{2,p}\left(J;{L}^{q}\left({\mathscr{F}}))\cap {H}_{\delta }^{1,p}\left(J;{H}^{2,q}\left({\mathscr{F}}))}\le C(| {e}^{{\varepsilon }_{0}t}{\partial }_{t}^{2}h{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}h{| }_{{H}_{\delta }^{1,p}\left(J;{H}^{2,q}\left({\mathscr{F}}))}).From (4.1)5{\left(4.1)}_{5}, we observe that ∂t2h−(P0G∂tu)⋅νG=∂tfhon G.{\partial }_{t}^{2}h-\left({P}_{0}^{{\mathscr{G}}}{\partial }_{t}u)\cdot {\nu }_{{\mathscr{G}}}={\partial }_{t}{f}_{h}\hspace{1.0em}\hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.Hence, we obtain ∣eε0t∂t2h∣Lδp(J;Lq(F))≤C(∣eε0t(P0G∂tu)⋅νG∣Lδp(J;Lq(F))+∣eε0t∂tfh∣Lδp(J;Lq(F)))≤C(∣eε0tu∣E1,δ(J;F)+∣eε0tfh∣F4,δ(J;G)),\begin{array}{rcl}| {e}^{{\varepsilon }_{0}t}{\partial }_{t}^{2}h{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}& \le & C(| {e}^{{\varepsilon }_{0}t}\left({P}_{0}^{{\mathscr{G}}}{\partial }_{t}u)\cdot {\nu }_{{\mathscr{G}}}{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}{\partial }_{t}{f}_{h}{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))})\\ & \le & C(| {e}^{{\varepsilon }_{0}t}u{| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})}+| {e}^{{\varepsilon }_{0}t}{f}_{h}{| }_{{{\mathbb{F}}}_{4,\delta }\left(J;{\mathscr{G}})}),\end{array}which implies ∣eε0th∣Fp,q,δ2−1/q(J;Lq(G))≤C(∣eε0t(u,h)∣E1,δ(J;F)×E4,δ(J;G)+∣eε0tfh∣F4,δ(J;G)).| {e}^{{\varepsilon }_{0}t}h{| }_{{F}_{p,q,\delta }^{2-1\text{/}q}\left(J;{L}^{q}\left({\mathscr{G}}))}\le C(| {e}^{{\varepsilon }_{0}t}\left(u,h){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}{f}_{h}{| }_{{{\mathbb{F}}}_{4,\delta }\left(J;{\mathscr{G}})}).It follows from (5.4) and (5.5) that ∫0T(eε0s∣(u∗(⋅,s),1)F∣)pds1/p+∫0T(eε0s∣(u∗(⋅,s),e3×y)F∣)pds1/p+∑α=1,2∫0Teε0s(u∗(⋅,s),eα×y)F−ω∫Gh∗(⋅,s)yαy3dGpds1/p≤C∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)},∑m=14∫0T(eε0s∣(h∗(⋅,s),φm)G∣)pds1/p≤C∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p+∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F),\begin{array}{l}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left({u}_{\ast }\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\le C\left\{{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right.\\ \hspace{1.0em}\hspace{1.0em}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}+| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}\},\\ \mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({h}_{\ast }\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\le C\left\{\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}\right\},\end{array}where CCdepends on ω\omega but is independent of TT. This completes the proof of Theorem 5.1.6The nonlinear problem6.1Local existenceWe now show the local existence result for given initial data (v0,Γ0)∈Bq,p2(δ−1/p)(Ω(0))3×Bq,p2+δ−1/p−1/q(G),\left({v}_{0},{\Gamma }_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left(\Omega \left(0))}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}),which are subject to the compatibility conditions the compatibility conditions divv0=0in Ω(0),PΓ0[μ(∇v0+[∇v0]⊤)]=0on Γ0{\rm{div}}\hspace{0.33em}{v}_{0}=0\hspace{1.0em}\hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(0)\text{},\hspace{1.0em}{{\mathcal{P}}}_{{\Gamma }_{0}}\left[\mu \left(\nabla {v}_{0}+{\left[\nabla {v}_{0}]}^{\top })]=0\hspace{1.0em}\hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\Gamma }_{0}\text{}and condition (1.4). We define the nonlinear mapping N=(N1,N2,N3,N4,N5,N6)N=\left({N}_{1},{N}_{2},{N}_{3},{N}_{4},{N}_{5},{N}_{6})by N1≔Fu(u,q,h)N2≔Gd(u,h)=divGdiv(u,h),N3≔Guτ(u,h)N4≔Guv(u,h)+G0(h)N5≔Fh(u,h)+F(u,h)\begin{array}{rcl}{N}_{1}& := & {F}_{u}\left(u,q,h)\hspace{1.0em}{N}_{2}:= {G}_{d}\left(u,h)={\rm{div}}\hspace{0.33em}{G}_{{\rm{div}}}\left(u,h),\\ {N}_{3}& := & {G}_{u\tau }\left(u,h)\hspace{1.0em}{N}_{4}:= {G}_{uv}\left(u,h)+{G}_{0}\left(h)\hspace{1.0em}{N}_{5}:= {F}_{h}\left(u,h)+F\left(u,h)\end{array}respectively. We set UT≔{z=(u,q,TrG[q],h)∈Eδ(J;F):∣h∣L∞(G×J)<ε}{{\mathbb{U}}}_{T}:= \left\{z=\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\hspace{0.33em}:\hspace{0.33em}| h{| }_{{L}^{\infty }\left({\mathscr{G}}\times J)}\lt \varepsilon \right\}with J=(0,T)J=\left(0,T), T>0T\gt 0, and 0<ε<10\lt \varepsilon \lt 1. Then we have the following proposition.Proposition 6.1Let (p,q,δ)\left(p,q,\delta )satisfy (1.13). Then it holds(1)NNis a real analytic mapping from UT{{\mathbb{U}}}_{T}to Fδ(J;F){{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})and N(0)=DN(0)=0N\left(0)=DN\left(0)=0.(2)DN(z)∈ℒ(UT,Fδ(J;F))DN\left(z)\in {\mathcal{ {\mathcal L} }}\left({{\mathbb{U}}}_{T},{{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})).Here, DNDNrepresents the Fréchet derivative of NN.ProofSince (p,q,δ)\left(p,q,\delta )satisfies (1.13), we have the following assertions, see [39, Lem. 5.3]: (i)E1,δ(J;F)↪BUC1(J;BUC(F)){{\mathbb{E}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{1}\left(J;\hspace{0.33em}{\rm{BUC}}\left({\mathscr{F}})).(ii)E3,δ(J;G)↪BUC(J;BUC(G)){{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{\rm{BUC}}\left(J;\hspace{0.33em}{\rm{BUC}}\left({\mathscr{G}})).(iii)E4,δ(J;G)↪BUC1(J;BUC1(G))∩BUC(J;BUC2(G)){{\mathbb{E}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{1}\left(J;\hspace{0.33em}{{\rm{BUC}}}^{1}\left({\mathscr{G}}))\cap {\rm{BUC}}\left(J;\hspace{0.33em}{{\rm{BUC}}}^{2}\left({\mathscr{G}})).(iv)E3,δ(J;G){{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})and F4,δ(J;G){{\mathbb{F}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})are multiplication algebras.Here, in assertions (i)–(iii), the embedding constants are independent of T>0T\gt 0if the time traces vanish at t=0t=0. The polynomial structure of the nonlinearity NNwith respect to uu, qq, and hhgives mapping properties for NN. This completes the proof.□Using the aforementioned proposition, we can obtain the local existence of classical solution to (1.1). Since the proof is standard (cf. [39, Sec. 5]), we may omit the detail.Theorem 6.2Let T>0T\gt 0be a given constant. Assume conditions (1.13) hold. Then there exists a constant ε=ε(T)>0\varepsilon =\varepsilon \left(T)\gt 0such that for arbitrary initial data (u0,η0)∈Bq,p2(δ−1/p)(F)3×Bq,p2+δ−1/p−1/q(G)\left({u}_{0},{\eta }_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfying the compatibility conditions(6.1)divu0=Gd(u0,h0)=divGdiv(u0,h0),inF,PF[μ(∇u0+[∇u0]⊤)]=Guτ(u0,h0),onF,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{\rm{div}}\hspace{0.33em}{u}_{0}={G}_{d}\left({u}_{0},{h}_{0})={\rm{div}}\hspace{0.33em}{G}_{{\rm{div}}}\left({u}_{0},{h}_{0}),& {in}\hspace{0.33em}{\mathscr{F}},\\ {{\mathcal{P}}}_{{\mathscr{F}}}\left[\mu \left(\nabla {u}_{0}+{\left[\nabla {u}_{0}]}^{\top })]={G}_{u\tau }\left({u}_{0},{h}_{0}),& {on}\hspace{0.33em}{\mathscr{F}},\end{array}\right.and the smallness condition∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(F)≤ε,| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{F}})}\le \varepsilon ,the transformed problem (3.7) has a unique solution (u,q,TrG[q],η)∈Eδ((0,T);F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta )\in {{\mathbb{E}}}_{\delta }\left(\left(0,T);\hspace{0.33em}{\mathscr{F}}). Furthermore, the solution is indeed real analytic in F×(0,T){\mathscr{F}}\times \left(0,T)and, especially, ℳ≔⋃t∈(0,T)(Γ(t)×{t}){\mathcal{ {\mathcal M} }}:= {\bigcup }_{t\in \left(0,T)}\left(\Gamma \left(t)\times \left\{t\right\})is a real analytic manifold.Remark 6.3To remove the smallness condition on the initial velocity field v0{v}_{0}, one has to consider the modified term Fh(u,h)+b⋅∇Gh{F}_{h}\left(u,h)+b\cdot {\nabla }_{{\mathscr{G}}}hinstead of Fh(u,h){F}_{h}\left(u,h), where b∈F4,δ(J;G)3b\in {{\mathbb{F}}}_{4,\delta }{\left(J;{\mathscr{G}})}^{3}is taken such that b(0)=TrG[PGu0]b\left(0)={{\rm{Tr}}}_{{\mathscr{G}}}\left[{{\mathcal{P}}}_{{\mathscr{G}}}{u}_{0}]. In fact, Fh(u,h){F}_{h}\left(u,h)cannot be small in the norm of F5,δ(J;G){{\mathbb{F}}}_{5,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})even if ∇G∣h∣L∞(G){\nabla }_{{\mathscr{G}}}| h{| }_{{L}^{\infty }\left({\mathscr{G}})}is small. However, to make our discussion simple, we keep the smallness assumption on v0{v}_{0}. Notice that the local existence result for arbitrary large initial velocity already obtained by Shibata [24], and see also [27, Thm. 3.6.1].6.2Global existence and convergenceFinally, we prove Theorem 1.2. In the following, we suppose that the initial data (u0,η0)∈Bq,p2(δ−1/p)(F)3×Bq,p2+δ−1/p−1/q(G)\left({u}_{0},{\eta }_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfies the smallness condition: (6.2)∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(F)≤ε| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{F}})}\le \varepsilon with some small ε>0\varepsilon \gt 0as well as the compatibility conditions (6.1). Since we will choose ε\varepsilon small eventually, we may suppose 0<ε<10\lt \varepsilon \lt 1. By Theorem 6.2, for given T0>0{T}_{0}\gt 0, there exists ε∗>0{\varepsilon }_{\ast }\gt 0such that (3.7) admits a unique solution (u,q,TrG[q],h)∈Eδ((0,T0);F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left(\left(0,{T}_{0});\hspace{0.33em}{\mathscr{F}}). In the following, we may assume ε<ε∗\varepsilon \lt {\varepsilon }_{\ast }. We further assume that the system (3.7) admits a solution (u,q,TrG[q],h)∈Eδ(J0;F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left({J}_{0};\hspace{0.33em}{\mathscr{F}})with J0=(0,T0){J}_{0}=\left(0,{T}_{0}). We shall show that the solution (u,q,TrG[q],h)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)can be prolong to the time interval R+{{\mathbb{R}}}_{+}. To this end, it suffices to verify the a priori estimate (6.3)∣eε0t(u,q,TrG[q],h)∣Eδ(0,T;F)≤C(∣(u0,η0)∣Bq,p2(δ−1/p)(F)×Bq,p2+δ−1/p−1/q(G)+∣eε0t(u,q,TrG[q],h)∣Eδ(0,T;F)2)| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;{\mathscr{F}})}\le C(| \left({u}_{0},{\eta }_{0}){| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})\times {B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2})for any T∈(0,T0]T\in \left(0,{T}_{0}], where a constant CCis independent of ε\varepsilon , TT, and T0{T}_{0}. Here, ε0{\varepsilon }_{0}is the same constant as in Theorem 5.1. In fact, combining the local existence result and the a priori estimate (6.3), a standard bootstrap argument implies the desired result.From Theorem 5.1 and Proposition 6.1, we easily find that (u,q,TrG[q],h)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)enjoys the estimate ∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)≤C[∣(u0,η0)∣Bq,p2(δ−1/p)(F)×Bq,p2+δ−1/p−1/q(G)+∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2+∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p.\begin{array}{l}| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;{\mathscr{F}})}\\ \hspace{1.0em}\le C{[}| \left({u}_{0},{\eta }_{0}){| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})\times {B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2}\\ \hspace{2.0em}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}\left.+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right].\end{array}A similar argument given in [26, Sec. 6] gives (6.4)∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p≤C∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2.\begin{array}{l}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\le C| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2}.\end{array}In fact, by [32, Sec. 2], it follows 0=∫Ω˜(t)dz−∫Fdy=∫Gh−h22HG+h33KGdG,0=∫Ω˜(t)zℓdz−∫Fyℓdy=∫Gh−h22HG+h33KGyℓ+νG(ℓ)h22−h33HG+h44KGdG,\begin{array}{rcl}0& =& \mathop{\displaystyle \int }\limits_{\widetilde{\Omega }\left(t)}{\rm{d}}z-\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\rm{d}}y=\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\left(h-\frac{{h}^{2}}{2}{{\mathscr{H}}}_{{\mathscr{G}}}+\frac{{h}^{3}}{3}{{\mathscr{K}}}_{{\mathscr{G}}}\right){\rm{d}}{\mathscr{G}},\\ 0& =& \mathop{\displaystyle \int }\limits_{\widetilde{\Omega }\left(t)}{z}_{\ell }{\rm{d}}z-\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{y}_{\ell }{\rm{d}}y=\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\left[\left(h-\frac{{h}^{2}}{2}{{\mathscr{H}}}_{{\mathscr{G}}}+\frac{{h}^{3}}{3}{{\mathscr{K}}}_{{\mathscr{G}}}\right){y}_{\ell }+{\nu }_{{\mathscr{G}}}^{\left(\ell )}\left(\frac{{h}^{2}}{2}-\frac{{h}^{3}}{3}{{\mathscr{H}}}_{{\mathscr{G}}}+\frac{{h}^{4}}{4}{{\mathscr{K}}}_{{\mathscr{G}}}\right)\right]{\rm{d}}{\mathscr{G}},\end{array}which gives the estimate (6.5)∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p≤C∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2,\mathop{\sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\le C| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2},cf., [26, Sec. 6]. Next, it follows from (1.4)2{\left(1.4)}_{2}and (1.6) that (v,eℓ×x)Ω(t)=(e3×y,eℓ×y)F=ωδℓ,3∫F∣y′∣2dy=−γδℓ,3,ℓ=1,2,3.{\left(v,{e}_{\ell }\times x)}_{\Omega \left(t)}={\left({e}_{3}\times y,{e}_{\ell }\times y)}_{{\mathscr{F}}}=\omega {\delta }_{\ell ,3}\mathop{\int }\limits_{{\mathscr{F}}}| y^{\prime} {| }^{2}{\rm{d}}y=-\gamma {\delta }_{\ell ,3},\hspace{1.0em}\ell =1,2,3.Hence, according to the transform explained in Section 3, we see that (6.6)(V˜,eℓ×z)Ω˜(t)+ω(e3×z,eℓ×z)Ω˜(t)=−γδℓ,3,ℓ=1,2,3.{\left(\widetilde{V},{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}+\omega {\left({e}_{3}\times z,{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}=-\gamma {\delta }_{\ell ,3},\hspace{1.0em}\ell =1,2,3.As J(h){\mathsf{J}}\left(h)describes the Jacobian of the transform Ξh{\Xi }_{h}, which is introduced in Section 3, we have (V˜,eℓ×z)Ω˜(t)=∫Fu⋅(eℓ×Ξh)J(h)dy=(u,eℓ×y)F+(u,eℓ×ξh)F+J0(h)(u,eℓ×Ξh)F{\left(\widetilde{V},{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}=\mathop{\int }\limits_{{\mathscr{F}}}u\cdot \left({e}_{\ell }\times {\Xi }_{h}){\mathsf{J}}\left(h){\rm{d}}y={\left(u,{e}_{\ell }\times y)}_{{\mathscr{F}}}+{\left(u,{e}_{\ell }\times {\xi }_{h})}_{{\mathscr{F}}}+{{\mathsf{J}}}_{0}\left(h){\left(u,{e}_{\ell }\times {\Xi }_{h})}_{{\mathscr{F}}}with ℓ=1,2,3\ell =1,2,3. Following [32, Sect. 2], for y∈Gy\in {\mathscr{G}}, we introduce J^(y,h)\widehat{{\mathsf{J}}}(y,h)defined by J^(y,h)≔∑i,j=13νG(i)(y)νG(j)(y)J(h)(I−[M1(h)]⊤)i,j,\widehat{{\mathsf{J}}}(y,h):= \mathop{\sum }\limits_{i,j=1}^{3}{\nu }_{{\mathscr{G}}}^{\left(i)}(y){\nu }_{{\mathscr{G}}}^{\left(j)}(y){\mathsf{J}}\left(h){(I-{\left[{M}_{1}\left(h)]}^{\top })}_{i,j},where M1{M}_{1}is the matrix given in Section 3. Then, by [32, p. 1772], for ℓ=1,2,3\ell =1,2,3, we may write (e3×z,eℓ×z)Ω˜(t)=(e3×y,eℓ×y)F+∫01∫G((e3×Ξrh)⋅(eℓ×Ξrh))hJ^(y,rh)dGdr,{\left({e}_{3}\times z,{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}={\left({e}_{3}\times y,{e}_{\ell }\times y)}_{{\mathscr{F}}}+\underset{0}{\overset{1}{\int }}\left(\mathop{\int }\limits_{{\mathscr{G}}}(\left({e}_{3}\times {\Xi }_{rh})\cdot \left({e}_{\ell }\times {\Xi }_{rh}))h\widehat{{\mathsf{J}}}(y,rh){\rm{d}}{\mathscr{G}}\right){\rm{d}}r,which yields the expression ω(e3×z,eℓ×z)Ω˜(t)=−γδℓ,3−ω∫Ghyℓy3dG+ωN˜ℓ(y,h).\omega {\left({e}_{3}\times z,{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}=-\gamma {\delta }_{\ell ,3}-\omega \mathop{\int }\limits_{{\mathscr{G}}}h{y}_{\ell }{y}_{3}{\rm{d}}{\mathscr{G}}+\omega {\widetilde{N}}_{\ell }(y,h).Here, N˜ℓ(y,h){\widetilde{N}}_{\ell }(y,h)is a nonlinear term that is given by N˜ℓ(y,h)=∫Ghyℓy3dG+∫01∫G((e3×Ξrh)⋅(eℓ×Ξrh))hJ^(y,rh)dGdr.{\widetilde{N}}_{\ell }(y,h)=\mathop{\int }\limits_{{\mathscr{G}}}h{y}_{\ell }{y}_{3}{\rm{d}}{\mathscr{G}}+\underset{0}{\overset{1}{\int }}\left(\mathop{\int }\limits_{{\mathscr{G}}}(\left({e}_{3}\times {\Xi }_{rh})\cdot \left({e}_{\ell }\times {\Xi }_{rh}))h\widehat{{\mathsf{J}}}(y,rh){\rm{d}}{\mathscr{G}}\right){\rm{d}}r.Hence, (6.6) turns into (u,eℓ×y)F−ω∫Ghyℓy3dG=−(u,eℓ×ξh)F−J0(h)(u,eℓ×Ξh)F−ωN˜ℓ(y,h),{\left(u,{e}_{\ell }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}h{y}_{\ell }{y}_{3}{\rm{d}}{\mathscr{G}}=-{\left(u,{e}_{\ell }\times {\xi }_{h})}_{{\mathscr{F}}}-{{\mathsf{J}}}_{0}\left(h){\left(u,{e}_{\ell }\times {\Xi }_{h})}_{{\mathscr{F}}}-\omega {\widetilde{N}}_{\ell }(y,h),where the right-hand side is nonlinear. Thanks to [39, Lem. 5.3], we have the estimate (6.7)∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p≤C∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2.{\left(\underset{0}{\overset{T}{\int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\le C| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2}.Combining (6.5) and (6.7), we observe (6.4). Thus, we obtain (6.3).Finally, we deal with the original problem (1.1). Notice that the compatibility condition (1.14) is valid if and only if (6.1) is satisfied. Define Ξh0(y)≔y+ξh0(y){\Xi }_{{h}_{0}}(y):= y+{\xi }_{{h}_{0}}(y)with replacing hhby h0{h}_{0}in the definition of ξh{\xi }_{h}. Then, the mapping Ξh0{\Xi }_{{h}_{0}}defines a C2{C}^{2}-diffeomorphism from F{\mathscr{F}}onto Ω(0)\Omega \left(0)with inverse Ξh0−1{\Xi }_{{h}_{0}}^{-1}. This gives the existence of a constant Ch0{C}_{{h}_{0}}depending on h0{h}_{0}such that Ch0−1∣v0−v∞∣Bq,p2(δ−1/p)(Ω(0))≤∣u0∣Bq,p2(δ−1/p)(F)≤Ch0∣v0−v∞∣Bq,p2(δ−1/p)(Ω(0)).{C}_{{h}_{0}}^{-1}| {v}_{0}-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(0))}\le | {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}\le {C}_{{h}_{0}}| {v}_{0}-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(0))}.Hence, there exists ε>0\varepsilon \gt 0such that the smallness condition of Theorem 1.2 yields (6.2). Recalling the discussion in Section 3, we see that there exists a unique global classical solution (v,π,Γ)\left(v,\pi ,\Gamma )to (1.1), especially, the unique global solution (v,π,Γ)\left(v,\pi ,\Gamma )to (1.1) is real analytic. Noting Bq,p2(δ−1/p)(Ω(t))≃Bq,p2(δ−1/p)(F){B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(t))\simeq {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left({\mathscr{F}}), we observe the asymptotic behavior of solutions. This completes the proof of Theorem 1.2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Nonlinear Analysis de Gruyter

Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension

Advances in Nonlinear Analysis , Volume 12 (1): 1 – Jan 1, 2023

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Publisher
de Gruyter
Copyright
© 2023 Keiichi Watanabe, published by De Gruyter
eISSN
2191-950X
DOI
10.1515/anona-2022-0279
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Abstract

1IntroductionThis article is concerned with the stability of equilibrium figure of uniformly rotating viscous incompressible fluid in R3{{\mathbb{R}}}^{3}with surface tension, where the equilibrium figure is rotationally symmetric about a certain axis. The fluid occupies a region Ω(t)\Omega \left(t)at time t≥0t\ge 0, which is surrounded by a free interface Γ(t)\Gamma \left(t). We denote the initial position of Γ(t)\Gamma \left(t)by Γ0{\Gamma }_{0}. Besides, we denote the velocity and pressure of the fluid by v(x,t)v\left(x,t)and π(x,t)\pi \left(x,t), respectively, and the unit outward normal on Γ(t)\Gamma \left(t)by νΓ{\nu }_{\Gamma }. The normal velocity and the doubled mean curvature of Γ(t)\Gamma \left(t)with respect to νΓ{\nu }_{\Gamma }are denoted by VΓ{V}_{\Gamma }and HΓ{{\mathscr{H}}}_{\Gamma }, respectively. Then, the motion of the fluid is governed by the following system: (1.1)∂tv+(v⋅∇)v−μΔv+∇π=0,in Ω(t),divv=0,in Ω(t),S(v,π)νΓ=σHΓνΓ,on Γ(t),VΓ=v⋅νΓ,on Γ(t),v(0)=v0,in Ω(0),Γ(0)=Γ0.\left\{\begin{array}{ll}{\partial }_{t}v+\left(v\cdot \nabla )v-\mu \Delta v+\nabla \pi =0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(t)\text{},\\ {\rm{div}}\hspace{0.33em}v=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(t)\text{},\\ S\left(v,\pi ){\nu }_{\Gamma }=\sigma {{\mathscr{H}}}_{\Gamma }{\nu }_{\Gamma },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\Gamma \left(t)\text{},\\ {V}_{\Gamma }=v\cdot {\nu }_{\Gamma },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\Gamma \left(t)\text{},\\ v\left(0)={v}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(0)\text{},\\ \Gamma \left(0)={\Gamma }_{0}.& \end{array}\right.In this article, Ω(0)\Omega \left(0)is assumed to be bounded. Here, S(v,π)S\left(v,\pi )is the stress tensor defined by S(v,π)≔μ(∇v+[∇v]⊤)−πIin Ω(t),S\left(v,\pi ):= \mu \left(\nabla v+{\left[\nabla v]}^{\top })-\pi I\hspace{1.0em}\hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(t)\text{},where μ\mu is a positive constant and M⊤{{\mathsf{M}}}^{\top }means the transpose of M{\mathsf{M}}. Without loss of generality, we may assume that external (atmospheric) pressure is zero in this article.It is well known that the Navier-Stokes equations admits the following stationary solution that corresponds to a rigid rotation: v∞(x)=ωe3×x,π∞(x)=ω22∣x′∣2+p0,{v}_{\infty }\left(x)=\omega {e}_{3}\times x,\hspace{1.0em}{\pi }_{\infty }\left(x)=\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}+{p}_{0},where ω∈R\omega \in {\mathbb{R}}describes a constant angular velocity and p0{p}_{0}is some constant, and we have set x′=(x1,x2,0)∈R3x^{\prime} =\left({x}_{1},{x}_{2},0)\in {{\mathbb{R}}}^{3}. Substituting (v∞,π∞)\left({v}_{\infty },{\pi }_{\infty })into the boundary condition (1.1)3{}_{3}, we obtain the equation for the doubled mean curvature HΓ∞{{\mathscr{H}}}_{{\Gamma }_{\infty }}of Γ∞{\Gamma }_{\infty }: (1.2)σHΓ∞+ω22∣x′∣2+p0=0onΓ∞≔∂Ω∞.\sigma {{\mathscr{H}}}_{{\Gamma }_{\infty }}+\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}+{p}_{0}=0\hspace{1.0em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\hspace{0.33em}{\Gamma }_{\infty }:= \partial {\Omega }_{\infty }.This defines the equilibrium figure Ω∞{\Omega }_{\infty }of the rotating liquid. Notice that, if Ω∞{\Omega }_{\infty }is rotationally symmetric about the x3{x}_{3}axis, the boundary condition (1.1)4{}_{4}is automatically satisfied. In the sequel, Ω∞{\Omega }_{\infty }is assumed to be rotationally symmetric with respect to the axis defined by e3{e}_{3}. Obviously, in the case ω=0\omega =0, we observe the classical Young-Laplace law σHΓ∞+p0=0\sigma {{\mathscr{H}}}_{{\Gamma }_{\infty }}+{p}_{0}=0. However, our interest here is to investigate the case ω≠0\omega \ne 0and, especially, characterize the stability of the equilibrium figure by means of the second variation of the energy functional instead of a restriction on the value of ω\omega .We note that a solution to (1.1) satisfies the following equalities: (1.3)∣Ω(t)∣=∣Ω(0)∣,∫Ω(t)v(x,t)dx=∫Ω(0)v0(x)dx,∫Ω(t)(v×x)dx=∫Ω(0)(v0×x)dx.| \Omega \left(t)| =| \Omega \left(0)| ,\hspace{1.0em}\mathop{\int }\limits_{\Omega \left(t)}v\left(x,t){\rm{d}}x=\mathop{\int }\limits_{\Omega \left(0)}{v}_{0}\left(x){\rm{d}}x,\hspace{1.0em}\mathop{\int }\limits_{\Omega \left(t)}\left(v\times x){\rm{d}}x=\mathop{\int }\limits_{\Omega \left(0)}\left({v}_{0}\times x){\rm{d}}x.By passing a uniformly moving coordinate system x^=x−V^0t\widehat{x}=x-{\widehat{V}}_{0}tand v^=v−V^0\widehat{v}=v-{\widehat{V}}_{0}, where V^0=∣Ω(0)∣−1∫Ω(0)v0dx{\widehat{V}}_{0}=| \Omega \left(0){| }^{-1}{\int }_{\Omega \left(0)}{v}_{0}{\rm{d}}x, and by rotating coordinate axes, we suppose that (1.4)∫Ω(t)v(x,t)dx=∫Ω(0)v0(x)dx=0,∫Ω(t)(v×x)dx=∫Ω(0)(v0×x)dx=γe3≔γ(0,0,1),∫Ω(0)xdx=0.\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{\Omega \left(t)}v\left(x,t){\rm{d}}x& =& \mathop{\displaystyle \int }\limits_{\Omega \left(0)}{v}_{0}\left(x){\rm{d}}x=0,\\ \mathop{\displaystyle \int }\limits_{\Omega \left(t)}\left(v\times x){\rm{d}}x& =& \mathop{\displaystyle \int }\limits_{\Omega \left(0)}\left({v}_{0}\times x){\rm{d}}x=\gamma {e}_{3}:= \gamma \left(0,0,1),\\ \mathop{\displaystyle \int }\limits_{\Omega \left(0)}x{\rm{d}}x& =& 0.\end{array}It follows form the Reynolds transport theorem that (1.4)3{\left(1.4)}_{3}gives (1.5)∫Ω(t)xdx=∫0tdds∫Ω(s)xdxds=∫0t∫Ω(s)vdxds=0,\mathop{\int }\limits_{\Omega \left(t)}x{\rm{d}}x=\underset{0}{\overset{t}{\int }}\frac{{\rm{d}}}{{\rm{d}}s}\left(\mathop{\int }\limits_{\Omega \left(s)}x{\rm{d}}x\right){\rm{d}}s=\underset{0}{\overset{t}{\int }}\left(\mathop{\int }\limits_{\Omega \left(s)}v{\rm{d}}x\right){\rm{d}}s=0,which means that the barycenter point of the domain Ω(t)\Omega \left(t)is always suited at the origin. Finally, to guarantee that (1.4)2{\left(1.4)}_{2}holds with v=v∞v={v}_{\infty }and Ω(t)=Ω∞\Omega \left(t)={\Omega }_{\infty }, the angular velocity ω\omega and the value γ\gamma should satisfy the relation (1.6)−ω∫Ω∞∣x′∣2dx=γ.-\omega \mathop{\int }\limits_{{\Omega }_{\infty }}| x^{\prime} {| }^{2}{\rm{d}}x=\gamma .Notice that the value ω\omega should be determined from a given quantity γ\gamma . Namely, Γ∞{\Gamma }_{\infty }is determined from γ\gamma , where Γ∞{\Gamma }_{\infty }is a smooth solution to (1.2) subject to (1.6). If ∣γ∣≪1| \gamma | \ll 1, there exists a unique Γ∞{\Gamma }_{\infty }satisfying (1.2) and (1.6), see Solonnikov [30, Thm. 5.1] and Watanabe [40, Prop. A.1]. Finally, the multiplication of (1.2) by x⋅νΓ∞x\cdot {\nu }_{{\Gamma }_{\infty }}and integration over Γ∞{\Gamma }_{\infty }leads to the expression for p0{p}_{0}, where νΓ∞{\nu }_{{\Gamma }_{\infty }}is the unit outward normal on Γ∞{\Gamma }_{\infty }. In fact, it follows from (1.2) that ∫Γ∞σHΓ∞x⋅νΓ∞dΓ∞+∫Γ∞ω22∣x′∣2x⋅νΓ∞dΓ∞+∫Γ∞p0x⋅νΓ∞dΓ∞=0.\mathop{\int }\limits_{{\Gamma }_{\infty }}\sigma {{\mathscr{H}}}_{{\Gamma }_{\infty }}x\cdot {\nu }_{{\Gamma }_{\infty }}{\rm{d}}{\Gamma }_{\infty }+\mathop{\int }\limits_{{\Gamma }_{\infty }}\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}x\cdot {\nu }_{{\Gamma }_{\infty }}{\rm{d}}{\Gamma }_{\infty }+\mathop{\int }\limits_{{\Gamma }_{\infty }}{p}_{0}x\cdot {\nu }_{{\Gamma }_{\infty }}{\rm{d}}{\Gamma }_{\infty }=0.The relation HΓ∞νΓ∞=ΔΓ∞x{{\mathscr{H}}}_{{\Gamma }_{\infty }}{\nu }_{{\Gamma }_{\infty }}={\Delta }_{{\Gamma }_{\infty }}x, x∈Γ∞x\in {\Gamma }_{\infty }, and the divergence theorem imply −σ∫Γ∞∣∇Γ∞x∣2dΓ∞+5ω22∫Ω∞∣x′∣2dx+3∣Ω∞∣p0=0.-\sigma \mathop{\int }\limits_{{\Gamma }_{\infty }}| {\nabla }_{{\Gamma }_{\infty }}x{| }^{2}{\rm{d}}{\Gamma }_{\infty }+\frac{5{\omega }^{2}}{2}\mathop{\int }\limits_{{\Omega }_{\infty }}| x^{\prime} {| }^{2}{\rm{d}}x+3| {\Omega }_{\infty }| {p}_{0}=0.Hence, we deduce that p0=2σ∣Γ∞∣3∣Ω∞∣+5γω6∣Ω∞∣.{p}_{0}=\frac{2\sigma | {\Gamma }_{\infty }| }{3| {\Omega }_{\infty }| }+\frac{5\gamma \omega }{6| {\Omega }_{\infty }| }.Notice that, by (1.6), we see that p0{p}_{0}is strictly positive.The free boundary problem of (1.1) is said to be finding a family of hypersurfaces {Γ(t)}t≥0{\left\{\Gamma \left(t)\right\}}_{t\ge 0}and appropriately smooth solutions vvand π\pi . Notice that finding a family of hypersurfaces {Γ(t)}t≥0{\left\{\Gamma \left(t)\right\}}_{t\ge 0}is equivalent to finding a family of {Ω(t)}t≥0{\left\{\Omega \left(t)\right\}}_{t\ge 0}. Since many authors have considered problems similar to (1.1) in various settings, we only mention the details of those papers that dealt with the effect of surface tension included on the free interface (the case σ>0\sigma \gt 0) and with assuming that Ω(0)\Omega \left(0)is bounded. For the case of surface tension on the free boundary, we refer the reader to papers [1,3,4, 5,11,17, 18,35,36, 37] that deal with the case where the initial domain Ω(0)\Omega \left(0)is an infinite layer of finite depth with a rigid bottom, see also a recent article by Saito and Shibata [23] that dealt with the case of the bottomless ocean. When Ω(0)\Omega \left(0)is bounded, the first contribution to the solvability of (1.1) traces back to a long series of papers by Solonnikov [29,30,31]. Specifically, Solonnikov investigated the problem in L2{L}^{2}regularity framework, i.e., he showed the local existence and uniqueness of solutions for (1.1) in Sobolev-Slobodetskiĭ spaces W22+α,1+α2{W}_{2}^{2+\alpha ,1+\frac{\alpha }{2}}with 1/2<α<11\hspace{0.1em}\text{/}\hspace{0.1em}2\lt \alpha \lt 1. To obtain the local-in-time solutions in Hölder and anisotropic Sobolev regularity frameworks, we refer to the works by Moglilevskiĭ and Solonnikov [16] and Shibata [24], respectively.To be precise, Shibata [24] obtained the local existence result for uniformly C3{C}^{3}-domains Ω(0)\Omega \left(0)such that the weak Dirichlet problem is uniquely solvable on W^01,q(Ω(0))≔{φ∈Llocq(Ω(0))∣∇φ∈Lq(Ω(0))3,φ∣Γ0=0}{\widehat{W}}_{0}^{1,q}\left(\Omega \left(0)):= \left\{\varphi \in {L}_{{\rm{loc}}}^{q}\left(\Omega \left(0))| \nabla \varphi \in {L}^{q}{\left(\Omega \left(0))}^{3},\varphi {| }_{{\Gamma }_{0}}=0\right\}, 1<q<∞1\lt q\lt \infty (cf. [27, Def. 3.2.5]).The unique global existence theorem in the L2{L}^{2}regularity frameworks was proved by Solonnikov [30], where the solution converges to a uniform rigid rotation of the liquid about a certain axis, provided that the initial velocity and the initial angular momentum (i.e., ∣γ∣| \gamma | ) are sufficiently small and Γ0{\Gamma }_{0}is sufficiently close to a sphere. The similar result was also established in Hölder regularity framework, see Padula and Solonnikov [19]. More recently, the author [40] extends the result obtained in [19] the class of anisotropic Sobolev spaces Wp,q2,1{W}_{p,q}^{2,1}with 2<p<∞2\lt p\lt \infty and 3<q<∞3\lt q\lt \infty satisfying 2/p+3/q<12\hspace{0.1em}\text{/}p+3\text{/}\hspace{0.1em}q\lt 1, which can also be regarded as an extension of Shibata [26] dealing with the case ω=0\omega =0. Notice that, in the previous studies [16,30,40], the equilibrium figure is uniquely determined by the constant γ\gamma . However, it was necessary to assume that ∣γ∣| \gamma | is small since this assumption yields the smallness of ∣ω∣| \omega | , so that we could find a unique smooth solution to equation (1.2) subject to (1.6) based on a standard contraction mapping theorem; see Solonnikov [30, Thm. 5.1] and Watanabe [40, Prop. A.1]. On the other hand, Solonnikov [34] showed that the smallness condition on ∣γ∣| \gamma | can be replaced by the condition of the positivity of the second variation of the functional Eω{E}_{\omega }given by (1.7)Eω(h)=∫Γ(t)σdΓ−∫Ω(t)ω22∣x′∣2dx−∫Ω(t)p0dx,{E}_{\omega }\left(h)=\mathop{\int }\limits_{\Gamma \left(t)}\sigma {\rm{d}}\Gamma -\mathop{\int }\limits_{\Omega \left(t)}\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}{\rm{d}}x-\mathop{\int }\limits_{\Omega \left(t)}{p}_{0}{\rm{d}}x,provided that there exists a smooth surface Γ∞{\Gamma }_{\infty }such that Γ∞{\Gamma }_{\infty }satisfies (1.2) and (1.6) and is rotationally symmetric about the x3{x}_{3}axis. In particular, he showed that the stability result can be obtained by the positivity of the second variation of the energy functional Eω(h){E}_{\omega }\left(h)within the Hölder regularity framework.The aim of this article is to extend the aforementioned Hölder regularity result obtained by Solonnikov [34, Thm. 2.1] in the Lp{L}^{p}-in-time and Lq{L}^{q}-in-space (Lp−Lq{L}^{p}-{L}^{q}) setting with 2<p<∞2\lt p\lt \infty and 3<q<∞3\lt q\lt \infty satisfying 2/p+3/q<12\hspace{0.1em}\text{/}p+3\text{/}\hspace{0.1em}q\lt 1, which provides an optimal regularity on the initial data. In particular, in contrast to Shibata [26], we investigate the stability of nontrivial stationary solutions (i.e., (v∞,π∞,Γ∞)\left({v}_{\infty },{\pi }_{\infty },{\Gamma }_{\infty })with ω≠0\omega \ne 0) to the three-dimensional Navier-Stokes equations with surface tension. To prove our main result, we transform the system (1.1) into a problem on a domain F{\mathscr{F}}surrounded by a fixed surface G{\mathscr{G}}. To observe smoothing of the unknown interface, we rely on the direct mapping method via the Hanzawa transform, i.e., we approximate the free surface Γ(t)\Gamma \left(t)by a real analytic hypersurface G{\mathscr{G}}, in terms of the Hausdorff distance of the second-order normal bundles being as small as we wish. As the transformed problem becomes a quasilinear parabolic type PDE with inhomogeneous boundary data, it is widely known that the key idea to show the local-in-time existence of regular solution is to use the maximal Lp−Lq{L}^{p}-{L}^{q}regularity result for an associated linearized problem. In this article, however, we will address the global existence issue, and hence, it is also required to derive some decay property of an associated linearized system – combining the local existence result and the decay property yields the global existence result due to a standard argument. To this end, we use the maximal Lp−Lq{L}^{p}-{L}^{q}regularity result for the shifted linearized system to rewrite the linearized system as an abstract evolution equation with homogeneous boundary data. Since the shifted linearized system admits a unique solution in the maximal Lp−Lq{L}^{p}-{L}^{q}regularity class that decays exponentially, it suffices to study the decay property for a solution to the abstract evolution equation. Although Solonnikov [34] established the decay estimate by energy estimates, his approach does not seem to be available in a general Lp−Lq{L}^{p}-{L}^{q}setting. To overcome this difficulty, we study the decay property of an analytic C0{C}_{0}-semigroup associated with the linearized system. On the basis of a spectral analysis of the corresponding linear operator Aq{{\mathcal{A}}}_{q}defined in X0=Jq(F)×Bq,q2−1/q(G){X}_{0}={J}_{q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), we will show that −A˜q≔−Aq∣X˜0-{\widetilde{{\mathcal{A}}}}_{q}:= -{{\mathcal{A}}}_{q}{| }_{{\widetilde{X}}_{0}}generates an analytic C0{C}_{0}-semigroup {e−A˜qt}t≥0{\left\{{e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}\right\}}_{t\ge 0}in X˜0{\widetilde{X}}_{0}, which is exponentially stable on some subspace X˜0{\widetilde{X}}_{0}of X0{X}_{0}, see Section 5. Here, Jq(F){J}_{q}\left({\mathscr{F}})and Bq,q2−1/q(G){B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})stand for the solenoidal space of (Lq(F))3{\left({L}^{q}\left({\mathscr{F}}))}^{3}and the inhomogeneous Besov spaces, respectively, and the space X˜0{\widetilde{X}}_{0}is defined as the set of all (f,g)∈X0(f,g)\in {X}_{0}that satisfies the following orthogonal conditions: ∫Ffdy=∫Ff⋅(e3×y)dy=0,∫Ff⋅(eα×y)dy=ω∫Ggyαy3dG,(α=1,2),∫GgdG=∫GgyℓdG=0,(ℓ=1,2,3),\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f{\rm{d}}y& =& \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{3}\times y){\rm{d}}y=0,\\ \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{\alpha }\times y){\rm{d}}y& =& \omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}g{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},& \left(\alpha =1,2),\\ \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}g{\rm{d}}{\mathscr{G}}& =& \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}g{y}_{\ell }{\rm{d}}{\mathscr{G}}=0,& \left(\ell =1,2,3),\end{array}which are similar to but essentially different from the orthogonal condition considered in Shibata [26]. Thanks to these orthogonal conditions, we will observe that the resolvent set of −A˜q-{\widetilde{{\mathcal{A}}}}_{q}contains the right half-plane C+≔{λ∈C:Reλ≥0}{{\mathbb{C}}}_{+}:= \left\{\lambda \in {\mathbb{C}}\hspace{0.33em}:\hspace{0.33em}{\rm{Re}}\hspace{0.33em}\lambda \ge 0\right\}including λ=0\lambda =0, which implies the exponential stability of the semigroup. Hence, our discussion generalizes the argument employed in Shibata [26] who deals with the case ω=0\omega =0, and see also Shibata and Shimizu [28] for the case σ=ω=0\sigma =\omega =0. Notice that, if γ=0\gamma =0(i.e., ω=0\omega =0), then the equilibrium surface becomes a sphere so that a natural choice of G{\mathscr{G}}is a sphere as well. If we choose G{\mathscr{G}}as a sphere, we obtain a nice spectral property of the Laplace-Beltrami operator defined on G{\mathscr{G}}, which arises from the surface tension, see [26, Lemm. 4.5] (cf. [22, Prop. 10.2.1]). In our case, however, it follows from ω≠0\omega \ne 0and (1.2) that the equilibrium surface is not sphere. In addition, since we do not impose the smallness condition on ∣γ∣| \gamma | (as well as ∣ω∣| \omega | ), the equilibrium surface cannot be understood as a normal perturbation of a sphere in general, which means that Shibata’s approach fails. If we assume that ∣γ∣| \gamma | is sufficiently small, we can recover Shibata’s argument because the equilibrium surface is given by a normal perturbation of a sphere, and see a recent contribution by the author [40]. Thus, we have to introduce another technique to handle the term arising from the surface tension. To this end, we introduce the quadratic form that determines from the functional Eω{E}_{\omega }given in (1.7).It should be noted that our result even in the case γ=ω=0\gamma =\omega =0refines Shibata’s result [26]. In fact, in view of the trace theory (cf. Denk et al. [9]), if we study the linearized problem, the boundary data have to be in the intersection space:(1.8)Fp,qs(0,T;Lq(G))∩Lp(0,T;Bq,qs/2(G)),0<s<1,{F}_{p,q}^{s}\left(0,T;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}^{p}\left(0,T;\hspace{0.33em}{B}_{q,q}^{s\hspace{0.1em}\text{/}\hspace{0.1em}2}\left({\mathscr{G}})),\hspace{1.0em}0\lt s\lt 1,where Fp,qs{F}_{p,q}^{s}denotes the vector-valued inhomogeneous Triebel-Lizorkin spaces. Hence, to find the strong solution to (1.1) via a contraction mapping principle, we have to estimate the nonlinear terms in this intersection space. However, in the argument in Shibata [26], the boundary data were not lying in (1.8), and hence, his result is not optimal in view of trace theory. Besides, Shibata [26] did not investigate that the height function hhadmits the higher regularity Fp,q2−1/q(J;Lq(G)){F}_{p,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}})), which implies that the free interface can be understood in the classical sense due to the embedding Fp,q2−1/q(0,T;Lq(G))∩H1,p(0,T;Bq,q2−1/q(G))∩Lp(0,T;Bq,q3−1/q(G))↪C([0,T];C2(G))∩C1([0,T];C1(G)).{F}_{p,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(0,T;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {H}^{1,p}\left(0,T;\hspace{0.33em}{B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\cap {L}^{p}\left(0,T;\hspace{0.33em}{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\hspace{0.33em}\hookrightarrow \hspace{0.33em}C\left(\left[0,T];\hspace{0.33em}{C}^{2}\left({\mathscr{G}}))\cap {C}^{1}\left(\left[0,T];\hspace{0.33em}{C}^{1}\left({\mathscr{G}})).To overcome these fallacious, we rely on the recent contributions established by the author [39,40], which were based on the studies of Lindemulder [13] and Meyries and Veraar [14,15]. Furthermore, it was not showed in the study by Shibata [25,26] that the solution can be understood in the classical sense, but the well-known parameter trick (cf. Prüss-Simonett [22, Ch. 9]) implies that the solution is indeed real analytic, jointly in time and space. This shows that the solutions to (1.1) are indeed classical.To explain our main result, we shall introduce the quadratic form that characterize the stability of the stationary solution to (1.1). As we will explain in the next section, the free surface Γ(t)\Gamma \left(t)can be approximated by a real analytic surface G{\mathscr{G}}in the sense that the Hausdorff distance of the second-order bundles of Γ(t)\Gamma \left(t)and G{\mathscr{G}}is as small as we wish. In this case, we can write (1.9)Γ(t)={p+h(p,t)νG(p):p∈G},Γ0={p+h0(p)νG(p):p∈G},\Gamma \left(t)=\left\{p+h\left(p,t){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {\mathscr{G}}\right\},\hspace{1em}{\Gamma }_{0}=\left\{p+{h}_{0}\left(p){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {\mathscr{G}}\right\},where hhis an unknown function but h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})is a given function. Here, we will prove that hhis indeed smooth function defined on G{\mathscr{G}}for each t>0t\gt 0. Since we seek global solutions that converges to an equilibrium, in the following, we may set G=Γ∞{\mathscr{G}}={\Gamma }_{\infty }. First, we observe that (1.2) is the Euler-Lagrange equation associated with Eω{E}_{\omega }. In fact, the first variation of Eω{E}_{\omega }at h=0h=0is given by (1.10)δ0Eω=−∫GσHGhdΓ−∫Gω22∣y′∣2hdG−∫Gp0hdG.{\delta }_{0}{E}_{\omega }=-\mathop{\int }\limits_{{\mathscr{G}}}\sigma {{\mathscr{H}}}_{{\mathscr{G}}}h{\rm{d}}\Gamma -\mathop{\int }\limits_{{\mathscr{G}}}\frac{{\omega }^{2}}{2}| y^{\prime} {| }^{2}h{\rm{d}}{\mathscr{G}}-\mathop{\int }\limits_{{\mathscr{G}}}{p}_{0}h{\rm{d}}{\mathscr{G}}.In addition, the second variation of Eω{E}_{\omega }at h=0h=0is given by (1.11)δ02Eω=−∫Gσ(hΔGh−2KGh2)dG−∫Gω22∂∂νG∣y′∣2−∣y′∣2HGh2dG+∫Gp0HGh2dG,{\delta }_{0}^{2}{E}_{\omega }=-\mathop{\int }\limits_{{\mathscr{G}}}\sigma (h{\Delta }_{{\mathscr{G}}}h-2{{\mathscr{K}}}_{{\mathscr{G}}}{h}^{2}){\rm{d}}{\mathscr{G}}-\mathop{\int }\limits_{{\mathscr{G}}}\frac{{\omega }^{2}}{2}\left(\frac{\partial }{\partial {\nu }_{{\mathscr{G}}}}| y^{\prime} {| }^{2}-| y^{\prime} {| }^{2}{{\mathscr{H}}}_{{\mathscr{G}}}\right){h}^{2}{\rm{d}}{\mathscr{G}}+\mathop{\int }\limits_{{\mathscr{G}}}{p}_{0}{{\mathscr{H}}}_{{\mathscr{G}}}{h}^{2}{\rm{d}}{\mathscr{G}},where KG{{\mathscr{K}}}_{{\mathscr{G}}}is the Gaussian curvature. We refer to [32, Sec. 2] for the derivations of (1.10) and (1.11), and see also [22, Ch. 2] for further geometric background. Recalling (1.2), we have σHG2+ω22∣y′∣2HG+p0HG=0onG,\sigma {{\mathscr{H}}}_{{\mathscr{G}}}^{2}+\frac{{\omega }^{2}}{2}| y^{\prime} {| }^{2}{{\mathscr{H}}}_{{\mathscr{G}}}+{p}_{0}{{\mathscr{H}}}_{{\mathscr{G}}}=0\hspace{1.0em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}},and hence, the second variation δ02Eω{\delta }_{0}^{2}{E}_{\omega }of Eω{E}_{\omega }can be rewritten as follows: δ02Eω=−∫GσhHG′(0)hdG−∫Gω22∂∂νG∣y′∣2h2dG≕∫GhℬGhdG,{\delta }_{0}^{2}{E}_{\omega }=-\mathop{\int }\limits_{{\mathscr{G}}}\sigma h{{\mathscr{H}}}_{{\mathscr{G}}}^{^{\prime} }\left(0)h{\rm{d}}{\mathscr{G}}-\mathop{\int }\limits_{{\mathscr{G}}}\frac{{\omega }^{2}}{2}\frac{\partial }{\partial {\nu }_{{\mathscr{G}}}}| y^{\prime} {| }^{2}{h}^{2}{\rm{d}}{\mathscr{G}}\hspace{0.33em}=: \hspace{0.33em}\mathop{\int }\limits_{{\mathscr{G}}}h{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}},where HG′(0){{\mathscr{H}}}_{{\mathscr{G}}}^{^{\prime} }\left(0)stands for the first variation of HG{{\mathscr{H}}}_{{\mathscr{G}}}at h=0h=0, since it holds ΔGh−2KGh=HG′(0)h−HG2h.{\Delta }_{{\mathscr{G}}}h-2{{\mathscr{K}}}_{{\mathscr{G}}}h={{\mathscr{H}}}_{{\mathscr{G}}}^{^{\prime} }\left(0)h-{{\mathscr{H}}}_{{\mathscr{G}}}^{2}h.Since δ0Eω=0{\delta }_{0}{E}_{\omega }=0, we observe that G{\mathscr{G}}is a minimal surface. We now normalize ℬGh{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}hby ℬ^Gh=ℬGh−1∣G∣∫GℬGhdG{\widehat{{\mathcal{ {\mathcal B} }}}}_{{\mathscr{G}}}h={{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h-\frac{1}{| {\mathscr{G}}| }\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}and define the quadratic form ΨG{\Psi }_{{\mathscr{G}}}by (1.12)ΨG(g,h)≔∫Ggℬ^GhdG{\Psi }_{{\mathscr{G}}}\left(g,h):= \mathop{\int }\limits_{{\mathscr{G}}}g{\widehat{{\mathcal{ {\mathcal B} }}}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}for g,h∈H2,2(G)g,h\in {H}^{2,2}\left({\mathscr{G}}). Then, we have δ02Eω=ΨG(h,h){\delta }_{0}^{2}{E}_{\omega }={\Psi }_{{\mathscr{G}}}\left(h,h)and ∫Gℬ^GhdG=0{\int }_{{\mathscr{G}}}{\widehat{{\mathcal{ {\mathcal B} }}}}_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}=0.The aim of this article is to show the stability of the stationary solution (v∞,π∞,G)\left({v}_{\infty },{\pi }_{\infty },{\mathscr{G}})to (1.1), provided that there exists a solution G{\mathscr{G}}to (1.2), and the quadratic form ΨG{\Psi }_{{\mathscr{G}}}satisfies the following assumption.Assumption 1.1The quadratic form ΨG(h,h){\Psi }_{{\mathscr{G}}}\left(h,h)is positive definite on L(0)2(G)≔h∈L2(G):∫GhdG=∫GhyjdG=0,j=1,2,3.,{L}_{\left(0)}^{2}\left({\mathscr{G}}):= \left\{h\in {L}^{2}\left({\mathscr{G}})\hspace{0.33em}:\hspace{0.33em}\mathop{\int }\limits_{{\mathscr{G}}}h{\rm{d}}{\mathscr{G}}=\mathop{\int }\limits_{{\mathscr{G}}}h{y}_{j}{\rm{d}}{\mathscr{G}}=0,\hspace{1.0em}j=1,2,3.\right\},that is, there exists a constant ccsuch that ∣ΨG(h,h)∣L2(G)≥c∣h∣L2(G)2| {\Psi }_{{\mathscr{G}}}\left(h,h){| }_{{L}^{2}\left({\mathscr{G}})}\ge c| h{| }_{{L}^{2}\left({\mathscr{G}})}^{2}holds for every h∈L(0)2(G)∩H2,2(G)h\in {L}_{\left(0)}^{2}\left({\mathscr{G}})\hspace{0.33em}\cap \hspace{0.33em}{H}^{2,2}\left({\mathscr{G}}).Our main result reads as follows.Theorem 1.2Suppose that Ω(0)\Omega \left(0)satisfies (1.4)3{\left(1.4)}_{3}. Assume that there exists a smooth solution Γ∞{\Gamma }_{\infty }to (1.2), which is rotationally symmetric about the x3{x}_{3}axis and that the quadratic form ΨG{\Psi }_{{\mathscr{G}}}satisfies Assumption 1.1. Let (v∞,π∞)\left({v}_{\infty },{\pi }_{\infty })be given by (1.3). If pp, qq, and δ\delta satisfy(1.13)2<p<∞,3<q<∞,1p+32q<δ−12≤12,2\lt p\lt \infty ,\hspace{1.0em}3\lt q\lt \infty ,\hspace{1.0em}\frac{1}{p}+\frac{3}{2q}\lt \delta -\frac{1}{2}\le \frac{1}{2},then an equilibrium (v∞,π∞,Γ∞)\left({v}_{\infty },{\pi }_{\infty },{\Gamma }_{\infty })is stable in the following sense: Let Γ0{\Gamma }_{0}be given by (1.9)2{\left(1.9)}_{2}with a given function h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}). There exists a positive constant ε>0\varepsilon \gt 0such that for all v0−v∞∈Bq,p2(δ−1/p)(Ω(0)){v}_{0}-{v}_{\infty }\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(0))and h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfying the smallness condition: ∣v0−v∞∣Bq,p2(δ−1/p)(Ω(0))+∣h0∣Bq,p2+δ−1/p−1/q(G)≤ε,| {v}_{0}-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(0))}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}\le \varepsilon ,the compatibility conditions: (1.14)divv0=0inΩ,PΓ0[μ(∇v0+[∇v0]⊤)]=0onΓ0{\rm{div}}\hspace{0.33em}{v}_{0}=0\hspace{1.0em}{in}\hspace{0.33em}\Omega ,\hspace{1.0em}{{\mathcal{P}}}_{{\Gamma }_{0}}\left[\mu \left(\nabla {v}_{0}+{\left[\nabla {v}_{0}]}^{\top })]=0\hspace{1.0em}{on}\hspace{0.33em}{\Gamma }_{0}and the conditions ∫Ω(0)v0(x)dx=0{\int }_{\Omega \left(0)}{v}_{0}\left(x){\rm{d}}x=0and ∫Ω(0)(v0×x)dx=γe3{\int }_{\Omega \left(0)}\left({v}_{0}\times x){\rm{d}}x=\gamma {e}_{3}with a constant γ∈R\gamma \in {\mathbb{R}}, there exists a unique global classical solution (v(t),π(t),Γ(t))\left(v\left(t),\pi \left(t),\Gamma \left(t))of problem (1.1). In particular, the set ⋃t∈(0,∞)(Γ(t)×{t}){\bigcup }_{t\in \left(0,\infty )}\left(\Gamma \left(t)\times \left\{t\right\})is real analytic manifold and the function (v,π):{(x,t)∈Ω(t)×(0,∞)}→R4\left(v,\pi ):\left\{\left(x,t)\in \Omega \left(t)\times \left(0,\infty )\right\}\to {{\mathbb{R}}}^{4}is real analytic. In addition, if Γ(t)\Gamma \left(t)is parameterized over G{\mathscr{G}}by means of a height function h(t)h\left(t), that is, if Γ(t)\Gamma \left(t)is given by (1.9)1{\left(1.9)}_{1}with an unknown function h, it holds∣v(t)−v∞∣Bq,p2(δ−1/p)(Ω(t))=O(e−ct)and∣h(t)∣Bq,p2+δ−1/p−1/q(G)=O(e−ct)ast→∞| v\left(t)-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(t))}=O\left({e}^{-ct})\hspace{1.0em}{and}\hspace{1.0em}| h\left(t){| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}=O\left({e}^{-ct})\hspace{1.0em}{as}\hspace{0.33em}t\to \infty with some positive constant c. Here, we have set PΓ0≔I−νΓ0⊗νΓ0{{\mathcal{P}}}_{{\Gamma }_{0}}:= I-{\nu }_{{\Gamma }_{0}}\otimes {\nu }_{{\Gamma }_{0}}.Remark 1.3The restriction (1.13) implies the embeddings Bq,p2(δ−1/p)(Ω(t))↪BUC1(Ω(t))andBq,p2+δ−1/p−1/q(G)↪BUC2(G).{B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(t))\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{1}\left(\Omega \left(t))\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{2}\left({\mathscr{G}}).Hence, we clearly observe that ∣v(t)−v∞∣C1(Ω(t))=O(e−ct)| v\left(t)-{v}_{\infty }{| }_{{C}^{1}\left(\Omega \left(t))}=O\left({e}^{-ct})and ∣h(t)∣C2(G)=O(e−ct)| h\left(t){| }_{{C}^{2}\left({\mathscr{G}})}=O\left({e}^{-ct})as t→∞t\to \infty . We also notice that the well known norm equivalence of the Hölder spaces Cs≃B∞,∞s{C}^{s}\simeq {B}_{\infty ,\infty }^{s}(s>0,s∉Ns\gt 0,s\notin {\mathbb{N}}) implies the embedding properties C2+α(Ω(0))≃B∞,∞2+α(Ω(0))↪Bq,p2(δ−1/p)(Ω(0)),α∈(0,1),C3+α(G)≃B∞,∞3+α(G)↪Bq,p2+δ−1/p−1/q(G),α∈(0,1),\begin{array}{rcl}{C}^{2+\alpha }\left(\Omega \left(0))& \simeq & {B}_{\infty ,\infty }^{2+\alpha }\left(\Omega \left(0))\hspace{0.33em}\hookrightarrow \hspace{0.33em}{B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(0)),\hspace{1.0em}\alpha \in \left(0,1),\\ {C}^{3+\alpha }\left({\mathscr{G}})& \simeq & {B}_{\infty ,\infty }^{3+\alpha }\left({\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}),\hspace{1.0em}\alpha \in \left(0,1),\end{array}which shows that the regularity of v0−v∞{v}_{0}-{v}_{\infty }and h0{h}_{0}in Theorem 1.2 are less than the assumption imposed in [34, Thm. 2.1]. Hence, Theorem 1.2 indeed improves the result due to Solonnikov [34, Thm. 2.1]. Furthermore, in contrast to [34, Thm. 2.1], solutions regularize and immediately become real analytic in space and time, which encodes typical parabolic behavior.Remark 1.4If the initial velocity v0{v}_{0}satisfies the orthogonal condition (1.4) with γ=0\gamma =0, Theorem 1.2 can be regarded as a generalization of Shibata [26].Remark 1.5Using elliptic integrals, the existence of the equilibrium surface Γ∞{\Gamma }_{\infty }that satisfies (1.6) may be shown and, especially, it is simply connected if the value of the nondimensional parameter h=ω2a38σ{\mathfrak{h}}=\frac{{\omega }^{2}{a}^{3}}{8\sigma }is strictly less than some value hmax≈2.32911{{\mathfrak{h}}}_{{\rm{\max }}}\approx 2.32911, and see Chandrasekhar [7] (cf. Appell [2, Ch. IX]). Here, aais the equatorial radius of the liquid drop and Γ∞{\Gamma }_{\infty }is symmetric with respect to an x1{x}_{1}-x2{x}_{2}plane. In the following, let us briefly explain the result presented by Chandrasekhar [7]. If h=0{\mathfrak{h}}=0(i.e., ω=0\omega =0), it is clear that Γ∞{\Gamma }_{\infty }is the sphere with the radius R>0R\gt 0; if 0<h≤10\lt {\mathfrak{h}}\le 1, then Γ∞{\Gamma }_{\infty }becomes an ellipsoid; if 1<h<hmax1\lt {\mathfrak{h}}\lt {{\mathfrak{h}}}_{{\rm{\max }}}, then a dimple appears. See also [7, Fig. 1] for a precise information. On the other hand, by [2, Ch. IX], if h≥hmax{\mathfrak{h}}\ge {{\mathfrak{h}}}_{{\rm{\max }}}, the equilibrium surface Γ∞{\Gamma }_{\infty }becomes a toroid (i.e., not simply connected), which replicates the classical experiment due to Plateau in the middle of the 19th century. In this article, it is not necessary to assume that the free boundary Γ(t)\Gamma \left(t)is simply connected, which is the same as Solonnikov [32], since we do not need to use polar coordinates to express the free boundary like Padula and Solonnikov [19]. Notice that Theorem 1.2 justifies the stability result, obtained by Brown and Scriven [6], for an axisymmetric equilibrium figure in the sense that we also consider the motion of the incompressible viscous fluid occupied inside the free boundary.In contrast to the previous article [40], Theorem 1.2 replaces the smallness condition on ∣γ∣| \gamma | by the condition of the positivity of δ02Eω=ΨG(h,h){\delta }_{0}^{2}{E}_{\omega }={\Psi }_{{\mathscr{G}}}\left(h,h), where it is assumed the existence of the equilibrium surface Γ∞{\Gamma }_{\infty }satisfying (1.6). However, Theorem 1.2 does not conflict with the result obtained in [40]. In fact, if ∣γ∣| \gamma | is suitably small, it follows from [40, Prop A.1] that there exists a unique Γ∞{\Gamma }_{\infty }satisfying (1.2) and (1.6) with Γ∞={p+h∞(p)νG(p):p∈SR},{\Gamma }_{\infty }=\left\{p+{h}_{\infty }\left(p){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {S}_{R}\right\},where ∣(h∞,∇h∞)∣L∞(SR)| \left({h}_{\infty },\nabla {h}_{\infty }){| }_{{L}^{\infty }\left({S}_{R})}are small. Here, SR{S}_{R}is a sphere centered at the origin with a radius R>0R\gt 0, and ∣(h∞,∇h∞)∣L∞(SR)| \left({h}_{\infty },\nabla {h}_{\infty }){| }_{{L}^{\infty }\left({S}_{R})}can be dominated by ∣γ∣| \gamma | , see [40, Prop. A.1]. In this case, setting p0=2σ/R{p}_{0}=2\sigma \hspace{0.1em}\text{/}\hspace{0.1em}R, the functional δ02Eω{\delta }_{0}^{2}{E}_{\omega }can be regarded as a perturbation from −∫SRσ(h¯ΔSRh¯−2KSRh¯2)dSR(h¯≔h−h∞),-\mathop{\int }\limits_{{S}_{R}}\sigma (\overline{h}{\Delta }_{{S}_{R}}\overline{h}-2{{\mathscr{K}}}_{{S}_{R}}{\overline{h}}^{2}){\rm{d}}{S}_{R}\hspace{1.0em}\left(\overline{h}:= h-{h}_{\infty }),which is positive definite on L(0)2(SR)∩H2,2(SR){L}_{\left(0)}^{2}\left({S}_{R})\cap {H}^{2,2}\left({S}_{R})(see [22, Prop. 10.2.1]). Hence, taking ∣γ∣| \gamma | as small as possible, we observe that the smallness of ∣γ∣| \gamma | implies the positivity of δ02Eω{\delta }_{0}^{2}{E}_{\omega }. Thus, we can conclude that Theorem 1.2 is an extension of the result obtained in the previous article. Accordingly, Theorem 1.2 includes all results obtained in [26,34,40].The rest of the article is folded as follows. In Section 2, we recall the notation of functional spaces and preliminary propositions used throughout this article. In Section 3, we transform the system (1.1) to a problem on a domain F{\mathscr{F}}surrounded by a fixed interface G{\mathscr{G}}in terms of the Hanzawa transform. Section 4 is devoted to showing that the principal part of the linearization has the property of maximal Lp−Lq{L}^{p}-{L}^{q}regularity. Then, some exponential decay property of the linearized system is proved in Section 5. Finally, the Section 6 presents the proof of the main result, Theorem 1.2.2Preliminaries2.1NotationsLet us fix the notations in this article. Let N{\mathbb{N}}be the set of all natural numbers and N0≔N∪{0}{{\mathbb{N}}}_{0}:= {\mathbb{N}}\cup \left\{0\right\}, and let R{\mathbb{R}}and C{\mathbb{C}}be, respectively, the set of all real numbers and the set of all complex numbers. Set R+≔{a∈R:a>0}{{\mathbb{R}}}_{+}:= \left\{a\in {\mathbb{R}}\hspace{0.33em}:\hspace{0.33em}a\gt 0\right\}. By C>0C\gt 0, we will often denote a generic constant that does not depend on the quantities at stake.2.2Functional spacesIn this subsection, we introduce functional spaces used throughout this article. Let p,q∈[1,∞]p,q\in \left[1,\infty ]. For any DDdomain of R3{{\mathbb{R}}}^{3}, we denote the standard K{\mathbb{K}}-valued Lebesgue spaces and Sobolev spaces on DDby Lq(D){L}^{q}\left(D)and Hm,q(D){H}^{m,q}\left(D), m∈Nm\in {\mathbb{N}}, respectively, where K∈{R,C}{\mathbb{K}}\in \left\{{\mathbb{R}},{\mathbb{C}}\right\}. The standard K{\mathbb{K}}-valued inhomogeneous Besov spaces on DDare denoted by Bp,qs(D){B}_{p,q}^{s}\left(D), s∈Rs\in {\mathbb{R}}. The homogeneous spaces H˙1,q(D){\dot{H}}^{1,q}\left(D)are given by H˙1,q(D)≔{w∈Lloc1(D):∇w∈Lq(D)}{\dot{H}}^{1,q}\left(D):= \left\{w\in {L}_{{\rm{loc}}}^{1}\left(D)\hspace{0.33em}:\hspace{0.33em}\nabla w\in {L}^{q}\left(D)\right\}for q∈(1,∞)q\in \left(1,\infty )and a domain D⊂R3D\subset {{\mathbb{R}}}^{3}, while the dual space of H˙1,q(D){\dot{H}}^{1,q}\left(D)is written by H˙−1,q′(D){\dot{H}}^{-1,q^{\prime} }\left(D)with the Hölder conjugate q′≔q/(q−1)q^{\prime} := q\hspace{0.1em}\text{/}\hspace{0.1em}\left(q-1)of qq.Let XXbe a Banach space. Then the mm-product space of XXis denoted by Xm{X}^{m}, while its norm is usually denoted by ∣⋅∣X| \cdot {| }_{X}instead of ∣⋅∣Xm| \cdot {| }_{{X}^{m}}when no confusion can arise.Let Banach spaces X1{X}_{1}and X2{X}_{2}the norm of X1∩X2{X}_{1}\cap {X}_{2}is denoted by ∣⋅∣X1∩X2≔∣⋅∣X1+∣⋅∣X2| \cdot {| }_{{X}_{1}\cap {X}_{2}}:= | \cdot {| }_{{X}_{1}}+| \cdot {| }_{{X}_{2}}, and ℒ(X1,X2){\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{2})stands for the Banach space of all bounded linear operators from X1{X}_{1}to X2{X}_{2}. We may write ℒ(X1){\mathcal{ {\mathcal L} }}\left({X}_{1})instead of ℒ(X1,X1){\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{1})to shorten the notation. The symbol Hol(Λ,ℒ(X1,X2)){\rm{Hol}}\hspace{0.33em}\left(\Lambda ,{\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{2}))represents the set of all ℒ(X1,X2){\mathcal{ {\mathcal L} }}\left({X}_{1},{X}_{2})-valued holomorphic functions defined on Λ⊂C\Lambda \subset {\mathbb{C}}. We set Σθ,z0≔{z∈C⧹{0}:∣argz∣≤π−θ,∣z∣≥z0}{\Sigma }_{\theta ,{z}_{0}}:= \left\{z\in {\mathbb{C}}\setminus \left\{0\right\}\hspace{0.33em}:\hspace{0.33em}| \arg z| \le \pi -\theta ,| z| \ge {z}_{0}\right\}with θ∈(0,π/2)\theta \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2)and z0>0{z}_{0}\gt 0.For I⊂RI\subset {\mathbb{R}}and p∈(1,∞]p\in \left(1,\infty ], let Lp(I;X){L}^{p}\left(I;\hspace{0.33em}X)and H1,p(I;X){H}^{1,p}\left(I;\hspace{0.33em}X)be the XX-valued Lebesgue spaces on IIand the XX-valued Sobolev spaces on II, respectively. For p∈(1,∞)p\in \left(1,\infty )and δ∈(1/p,1]\delta \in \left(1\hspace{0.1em}\text{/}\hspace{0.1em}p,1], we set Lδp(I;X)≔{f:I→X:t1−δf∈Lp(I;X)},Hδ1,p(I;X)≔{f∈Lδp(I;X)∩H1,1(I;X):∂tf∈Lδp(I;X)}.\begin{array}{rcl}{L}_{\delta }^{p}\left(I;\hspace{0.33em}X)& := & \{f:I\to X\hspace{0.33em}:\hspace{0.33em}{t}^{1-\delta }f\in {L}^{p}\left(I;\hspace{0.33em}X)\},\\ {H}_{\delta }^{1,p}\left(I;\hspace{0.33em}X)& := & \{f\in {L}_{\delta }^{p}\left(I;\hspace{0.33em}X)\cap {H}^{1,1}\left(I;\hspace{0.33em}X)\hspace{0.33em}:\hspace{0.33em}{\partial }_{t}f\in {L}_{\delta }^{p}\left(I;\hspace{0.33em}X)\}.\end{array}For p∈(1,∞)p\in \left(1,\infty ), q∈[1,∞]q\in \left[1,\infty ], and s∈Rs\in {\mathbb{R}}, the symbol Fp,q,δs(I;X){F}_{p,q,\delta }^{s}\left(I;\hspace{0.33em}X)stands for the XX-valued Triebel-Lizorkin spaces with the power weight ∣t∣p(1−δ)| t{| }^{p\left(1-\delta )}. In addition, the Banach space of all XX-valued bounded uniformly continuous functions on IIis denoted by BUC(I;X){\rm{BUC}}\left(I;\hspace{0.33em}X). Finally, BUCm(I;X){{\rm{BUC}}}^{m}\left(I;\hspace{0.33em}X)is the subset of BUC(I;X){\rm{BUC}}\left(I;\hspace{0.33em}X)that has bounded partial derivatives up to order m∈Nm\in {\mathbb{N}}. Here, BUC(D){\rm{BUC}}\left(D)and BUCm(D){{\rm{BUC}}}^{m}\left(D)are defined similarly as mentioned earlier. For further information on function spaces, we refer the reader to [22,38].2.3The space of data for divergence equationAs we will see in Section 2.4, in a transformed system, we have the divergence equation divu=gd{\rm{div}}\hspace{0.33em}u={g}_{d}in F{\mathscr{F}}. Hence, to deal with this equation, it is required to introduce a space DIq(F){{\rm{DI}}}_{q}\left({\mathscr{F}})that is the set of all gd∈H1,q(F){g}_{d}\in {H}^{1,q}\left({\mathscr{F}})such that there exists a solution g˜d∈Lq(F)3{\widetilde{g}}_{d}\in {L}^{q}{\left({\mathscr{F}})}^{3}to (gd,φ)F=−(g˜d,∇φ)F{\left({g}_{d},\varphi )}_{{\mathscr{F}}}=-{\left({\widetilde{g}}_{d},\nabla \varphi )}_{{\mathscr{F}}}for every φ∈H01,q′(F)≔{φ∈H1,q(F):φ∣G=0}\varphi \in {H}_{0}^{1,q^{\prime} }\left({\mathscr{F}}):= \left\{\varphi \in {H}^{1,q}\left({\mathscr{F}})\hspace{0.33em}:\hspace{0.33em}\varphi {| }_{{\mathscr{G}}}=0\right\}. Here and in the following, (⋅,⋅)F{\left(\cdot ,\cdot )}_{{\mathscr{F}}}denotes the L2{L}^{2}inner product in F{\mathscr{F}}. We now define G(gd)≔{g∗∈Lq(F)3:divg˜d=divg∗}{\mathsf{G}}\left({g}_{d}):= \left\{{g}_{\ast }\in {L}^{q}{\left({\mathscr{F}})}^{3}\hspace{0.33em}:\hspace{0.33em}{\rm{div}}\hspace{0.33em}{\widetilde{g}}_{d}={\rm{div}}\hspace{0.33em}{g}_{\ast }\right\}and denote the representative elements of G(gd){\mathsf{G}}\left({g}_{d})by [G(gd)]\left[{\mathsf{G}}\left({g}_{d})]. For brevity, and when there is no danger of confusion, we write G(gd){\mathsf{G}}\left({g}_{d})instead of [G(gd)]\left[{\mathsf{G}}\left({g}_{d})]. Here, it holds divG(gd)=gd{\rm{div}}\hspace{0.33em}{\mathsf{G}}\left({g}_{d})={g}_{d}in F{\mathscr{F}}for every gd∈DIq(F){g}_{d}\in {{\rm{DI}}}_{q}\left({\mathscr{F}}). Setting ∣gd∣DIq(F)≔∣gd∣H1,q(F)+infg∗∈G(gd)∣g∗∣Lq(F)forgd∈DIq(F),| {g}_{d}{| }_{{{\rm{DI}}}_{q}\left({\mathscr{F}})}:= | {g}_{d}{| }_{{H}^{1,q}\left({\mathscr{F}})}+\mathop{\inf }\limits_{{g}_{\ast }\in {\mathsf{G}}\left({g}_{d})}| {g}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}{g}_{d}\in {{\rm{DI}}}_{q}\left({\mathscr{F}}),we observe that DIq(F){{\rm{DI}}}_{q}\left({\mathscr{F}})is a Banach space endowed with the norm ∣⋅∣DIq(F)| \cdot {| }_{{{\rm{DI}}}_{q}\left({\mathscr{F}})}.2.4ℛ{\mathcal{ {\mathcal R} }}-bounded families of operatorsWe next recall the basic theory of the ℛ{\mathcal{ {\mathcal R} }}-boundedness of a family of operators, which will be used in Section 4. Here, we refer to [8] for the fundamental concept of ℛ{\mathcal{ {\mathcal R} }}-boundedness. In the following, the ℛ{\mathcal{ {\mathcal R} }}-bound of a family of operators T⊂ℒ(X,Y){\mathcal{T}}\subset {\mathcal{ {\mathcal L} }}\left(X,Y)is denoted by ℛX→Y{T}{{\mathcal{ {\mathcal R} }}}_{X\to Y}\left\{{\mathcal{T}}\right\}, where XXand YYare Banach spaces. If X=YX=Y, we ofren write ℛX{T}{{\mathcal{ {\mathcal R} }}}_{X}\left\{{\mathcal{T}}\right\}for short. The following result is a direct consequence of [8, Rem. 3.2 (4)].Proposition 2.1Let 1≤q<∞1\le q\lt \infty and G⊂R3G\subset {{\mathbb{R}}}^{3}be a domain. Let m(λ)m\left(\lambda )be a bounded function defined on a subset Λ\Lambda of C{\mathbb{C}}, and let Mm(λ){M}_{m}\left(\lambda )be a multiplication operator given by Mm(λ)f≔m(λ)f{M}_{m}\left(\lambda )f:= m\left(\lambda )ffor every f∈Lq(G)f\in {L}^{q}\left(G\right). Then it holdsℛLq(G)({Mm(λ):λ∈Λ})≤Kq2(supλ∈Λ∣m(λ)∣),{{\mathcal{ {\mathcal R} }}}_{{L}^{q}\left(G)}\left(\left\{{M}_{m}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in \Lambda \right\})\le {K}_{q}^{2}\left(\mathop{\sup }\limits_{\lambda \in \Lambda }| m\left(\lambda \right)| \right),where Kq>0{K}_{q}\gt 0is a constant appearing in the Khintchine inequality.3Reduction to a fixed reference surfaceIn this section, we transform problem (1.1) to one on the fixed spatial domain t≥0t\ge 0. Putting V≔v−v∞V:= v-{v}_{\infty }and P=π−π∞P=\pi -{\pi }_{\infty }, the problem of a stability of the equilibrium (v∞,π∞,Γ∞)\left({v}_{\infty },{\pi }_{\infty },{\Gamma }_{\infty })reduces to a free boundary problem for the perturbation (V,P)\left(V,P): (3.1)∂tV+(v∞⋅∇)V+(V⋅∇)v∞+(V⋅∇)V−μΔV+∇P=0,inΩ(t),divV=0,inΩ(t),S(V,P)νΓ=σHΓ+ω22∣x′∣2+p0νΓ,onΓ(t),VΓ=(v∞+V)⋅νΓ,onΓ(t),V(0)=v0−v∞,inΩ(0),Γ(0)=Γ0.\left\{\begin{array}{ll}{\partial }_{t}V+\left({v}_{\infty }\cdot \nabla )V+\left(V\cdot \nabla ){v}_{\infty }+\left(V\cdot \nabla )V-\mu \Delta V+\nabla P=0,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \left(t),\\ {\rm{div}}\hspace{0.33em}V=0,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\hspace{0.33em}\Omega \left(t),\\ S\left(V,P){\nu }_{\Gamma }=\left(\sigma {{\mathscr{H}}}_{\Gamma }+\frac{{\omega }^{2}}{2}| x^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{\Gamma },& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\Gamma \left(t),\\ {V}_{\Gamma }=\left({v}_{\infty }+V)\cdot {\nu }_{\Gamma },& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\Gamma \left(t),\\ V\left(0)={v}_{0}-{v}_{\infty },& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \left(0),\\ \Gamma \left(0)={\Gamma }_{0}.& \end{array}\right.In our analysis, it will be useful to eliminate the terms (v∞⋅∇)V\left({v}_{\infty }\cdot \nabla )Vand (V⋅∇)v∞\left(V\cdot \nabla ){v}_{\infty }. To this end, we introduce the coordinate system rotating about the x3{x}_{3}axis with a constant angular velocity ω∈R\omega \in {\mathbb{R}}, i.e., x=O(ωt)zx={\mathcal{O}}\left(\omega t)z. We now set V˜(z,t)≔O−1(ωt)V(O(ωt)z,t)andP˜(z,t)≔P(O(ωt)z,t)\widetilde{V}\left(z,t):= {{\mathcal{O}}}^{-1}\left(\omega t)V\left({\mathcal{O}}\left(\omega t)z,t)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\widetilde{P}\left(z,t):= P\left({\mathcal{O}}\left(\omega t)z,t)with O(θ)≔cosθ−sinθ0sinθcosθ0001.{\mathcal{O}}\left(\theta ):= \left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right).Then problem (3.1) can be rewritten as follows: (3.2)∂tV˜+(V˜⋅∇)V˜−μΔV˜+2ω(e3×V˜)+∇P˜=0,in Ω˜(t),divV˜=0,in Ω˜(t),S(V˜,P˜)νΓ˜=σHΓ˜+ω22∣z′∣2+p0νΓ˜,on Γ˜(t),VΓ˜=V˜⋅νΓ˜,on Γ˜(t),V˜(0)=v0−v∞≕V˜0,in Ω˜(0),Γ(0)=Γ0,\left\{\begin{array}{ll}{\partial }_{t}\widetilde{V}+\left(\widetilde{V}\cdot \nabla )\widetilde{V}-\mu \Delta \widetilde{V}+2\omega \left({e}_{3}\times \widetilde{V})+\nabla \widetilde{P}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\widetilde{\Omega }\left(t)\text{},\\ {\rm{div}}\hspace{0.33em}\widetilde{V}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\widetilde{\Omega }\hspace{0.33em}\left(t)\text{},\\ S\left(\widetilde{V},\widetilde{P}){\nu }_{\widetilde{\Gamma }}=\left(\sigma {{\mathscr{H}}}_{\widetilde{\Gamma }}+\frac{{\omega }^{2}}{2}| z^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{\widetilde{\Gamma }},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\widetilde{\Gamma }\hspace{0.33em}\left(t)\text{},\\ {V}_{\widetilde{\Gamma }}=\widetilde{V}\cdot {\nu }_{\widetilde{\Gamma }},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}\widetilde{\Gamma }\hspace{0.33em}\left(t)\text{},\\ \widetilde{V}\left(0)={v}_{0}-{v}_{\infty }\hspace{0.33em}=: \hspace{0.33em}{\widetilde{V}}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\widetilde{\Omega }\left(0)\text{},\\ \Gamma \left(0)={\Gamma }_{0},& \end{array}\right.where we have set Ω˜(t)=O−1(ωt)Ω(t)\widetilde{\Omega }\left(t)={{\mathcal{O}}}^{-1}\left(\omega t)\Omega \left(t)and Γ˜(t)=∂Ω˜(t)\widetilde{\Gamma }\left(t)=\partial \widetilde{\Omega }\left(t). Here, VΓ˜(⋅,t){V}_{\widetilde{\Gamma }}\left(\cdot ,t)and HΓ˜(⋅,t){{\mathscr{H}}}_{\widetilde{\Gamma }}\left(\cdot ,t)stand for the normal velocity and the doubled mean curvature of Γ˜(t)\widetilde{\Gamma }\left(t)with respect to the norm νΓ˜(⋅,t)=O−1(ωt)νΓ(⋅,t){\nu }_{\widetilde{\Gamma }}\left(\cdot ,t)={{\mathcal{O}}}^{-1}\left(\omega t){\nu }_{\Gamma }\left(\cdot ,t)to Γ˜(t)\widetilde{\Gamma }\left(t), respectively. For further details of the derivation of (3.2), the readers may consult the discussion in [34]. It follows from conditions (1.4)3{\left(1.4)}_{3}and (1.5) that ∫Ω˜(t)zdz=∫Ω˜(0)dz=0,\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}z{\rm{d}}z=\mathop{\int }\limits_{\widetilde{\Omega }\left(0)}{\rm{d}}z=0,i.e., the barycenter of the fluid is still located at the origin. In addition, condition (1.4)1{\left(1.4)}_{1}becomes ∫Ω˜(t)V˜(z,t)dz=0\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}\left(z,t){\rm{d}}z=0due to ∫Ω˜(t)zdz=0{\int }_{\widetilde{\Omega }\left(t)}z{\rm{d}}z=0. Notice that (3.2)4{\left(3.2)}_{4}is equivalent to VΓ˜(z,t)=V˜⋅νΓ˜−1∣F∣νΓ˜∫Ω˜(t)V˜(z,t)dz.{V}_{\widetilde{\Gamma }}\left(z,t)=\widetilde{V}\cdot {\nu }_{\widetilde{\Gamma }}-\frac{1}{| {\mathscr{F}}| }{\nu }_{\widetilde{\Gamma }}\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}\left(z,t){\rm{d}}z.We will see in Section 5 that this modification will be convenient.Next, we transform (3.2) to a system on a domain F{\mathscr{F}}surrounded by a fixed surface G{\mathscr{G}}via the Hanzawa transform, where the strategy is due to Köhne et al. [12, Sec. 2] (cf. Prüss and Siminett [22]). For the necessary geometric background, we refer to Chapter 2 in [22]. Recall that the second-order bundle of Γ˜\widetilde{\Gamma }is given by N2Γ˜≔{(p,νΓ˜(p),∇Γ˜νΓ˜(p)):p∈Γ˜},{{\mathcal{N}}}^{2}\widetilde{\Gamma }:= \left\{\left(p,{\nu }_{\widetilde{\Gamma }}\left(p),{\nabla }_{\widetilde{\Gamma }}{\nu }_{\widetilde{\Gamma }}\left(p))\hspace{0.33em}:\hspace{0.33em}p\in \widetilde{\Gamma }\right\},where νΓ˜{\nu }_{\widetilde{\Gamma }}is the surface gradient on Γ˜\widetilde{\Gamma }. In addition, let us write dH{d}_{H}to denote the Hausdorff distance between the closed subsets K1,K2⊂R3{K}_{1},{K}_{2}\subset {{\mathbb{R}}}^{3}, which is defined by dH(K1,K2)≔max(supa∈K1dist(a,K2),supb∈K2dist(b,K1)).{d}_{H}\left({K}_{1},{K}_{2}):= \max \left(\mathop{\sup }\limits_{a\in {K}_{1}}{\rm{dist}}\hspace{0.33em}\left(a,{K}_{2}),\mathop{\sup }\limits_{b\in {K}_{2}}{\rm{dist}}\hspace{0.33em}\left(b,{K}_{1})\right).Then, the unknown surface Γ˜\widetilde{\Gamma }can be approximated by a real analytic hypersurface G{\mathscr{G}}in the sense that dH(N2Γ˜,N2G){d}_{H}\left({{\mathcal{N}}}^{2}\widetilde{\Gamma },{{\mathcal{N}}}^{2}{\mathscr{G}})is as small as we wish, i.e., for each η>0\eta \gt 0, we can find a real analytic closed hypersurface G{\mathscr{G}}such that dH(N2Γ˜,N2G)≤η{d}_{H}\left({{\mathcal{N}}}^{2}\widetilde{\Gamma },{{\mathcal{N}}}^{2}{\mathscr{G}})\le \eta . Since it is well known that the hypersurface G{\mathscr{G}}admits a tubular neighborhood, there exists some positive constant d0{{\mathsf{d}}}_{0}such that the mapping Λ:G×(−d0,d0)→R3\Lambda :{\mathscr{G}}\times \left(-{{\mathsf{d}}}_{0},{{\mathsf{d}}}_{0})\to {{\mathbb{R}}}^{3}defined by Λ(p,r)≔p+rνG(p),p∈G,∣r∣<d0\Lambda \left(p,r):= p+r{\nu }_{{\mathscr{G}}}\left(p),\hspace{1.0em}p\in {\mathscr{G}},| r| \lt {{\mathsf{d}}}_{0}is a diffeomorphism from G×(−d0,d0){\mathscr{G}}\times \left(-{{\mathsf{d}}}_{0},{{\mathsf{d}}}_{0})onto R(Λ){\mathsf{R}}\left(\Lambda ). Here, R(Λ){\mathsf{R}}\left(\Lambda )denotes the range of Λ\Lambda . Then, the inverse Λ−1:R(Λ)→G×(−d0,d0){\Lambda }^{-1}:{\mathsf{R}}\left(\Lambda )\to {\mathscr{G}}\times \left(-{{\mathsf{d}}}_{0},{{\mathsf{d}}}_{0})is conveniently decomposed as Λ−1(y)=(ΠG(y),dG(y)){\Lambda }^{-1}(y)=\left({\Pi }_{{\mathscr{G}}}(y),{d}_{{\mathscr{G}}}(y))for y∈R(Λ)y\in {\mathsf{R}}\left(\Lambda ), where ΠG(y){\Pi }_{{\mathscr{G}}}(y)and dG(y){d}_{{\mathscr{G}}}(y)stand for the orthogonal projection of yyonto G{\mathscr{G}}and the signed distance from yyonto G{\mathscr{G}}; so ∣dG(y)∣=dist(y,G)| {d}_{{\mathscr{G}}}(y)| ={\rm{dist}}\hspace{0.33em}(y,{\mathscr{G}})and dG(y)<0{d}_{{\mathscr{G}}}(y)\lt 0if and only if y∈Fy\in {\mathscr{F}}. Noting the compactness of G{\mathscr{G}}, there exists a radius rG>0{r}_{{\mathscr{G}}}\gt 0such that for each point p∈Gp\in {\mathscr{G}}there are balls B(y,rG)⊂FB(y,{r}_{{\mathscr{G}}})\subset {\mathscr{F}}satisfying G∩B(y,rG)¯={p}{\mathscr{G}}\cap \overline{B(y,{r}_{{\mathscr{G}}})}=\left\{p\right\}. Choosing rG{r}_{{\mathscr{G}}}maximal, it holds rG>d0{r}_{{\mathscr{G}}}\gt {{\mathsf{d}}}_{0}. In the following, we fix d0=rG/2{{\mathsf{d}}}_{0}={r}_{{\mathscr{G}}}\hspace{0.1em}\text{/}\hspace{0.1em}2and d=d0/3{\mathsf{d}}={{\mathsf{d}}}_{0}\hspace{0.1em}\text{/}\hspace{0.1em}3.We write the derivatives of dG(y){d}_{{\mathscr{G}}}(y)and ΠG(y){\Pi }_{{\mathscr{G}}}(y)by ∇dG(y)=νG(ΠG(y)),DΠG(y)=M0(dG(y))PG(ΠG(y)),\nabla {d}_{{\mathscr{G}}}(y)={\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)),\hspace{1.0em}D{\Pi }_{{\mathscr{G}}}(y)={M}_{0}\left({d}_{{\mathscr{G}}}(y)){{\mathcal{P}}}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)),respectively. Here, M0(r)≔(I−rLG)−1{M}_{0}\left(r):= {\left(I-r{L}_{{\mathscr{G}}})}^{-1}is the Weingarten tensor LG≔−∇GνG{L}_{{\mathscr{G}}}:= -{\nabla }_{{\mathscr{G}}}{\nu }_{{\mathscr{G}}}and PG(p)=I−νG(p)⊗νG(p){{\mathcal{P}}}_{{\mathscr{G}}}\left(p)=I-{\nu }_{{\mathscr{G}}}\left(p)\otimes {\nu }_{{\mathscr{G}}}\left(p)represents the orthogonal projection onto the tangent space of G{\mathscr{G}}at p∈Gp\in {\mathscr{G}}. Here, it holds ∣M0(r)∣≤(1−r∣LG∣)≤3| {M}_{0}\left(r)| \le \left(1-r| {L}_{{\mathscr{G}}}| )\le 3for all ∣r∣≤2rG/3| r| \le 2{r}_{{\mathscr{G}}}\hspace{0.1em}\text{/}\hspace{0.1em}3. Using the mapping Λ\Lambda , we may approximate the unknown free surface Γ˜(t)\widetilde{\Gamma }\left(t)over G{\mathscr{G}}by means of a height function hhvia Γ˜(t)≔{p+h(p,t)νG(p):p∈G}\widetilde{\Gamma }\left(t):= \left\{p+h\left(p,t){\nu }_{{\mathscr{G}}}\left(p)\hspace{0.33em}:\hspace{0.33em}p\in {\mathscr{G}}\right\}for small t≥0t\ge 0, at least. We extend this diffeomorphism to all of F¯\overline{{\mathscr{F}}}and define the Hanzawa transform by Ξh(y,t)≔y+χdG(y)dh(ΠG(y),t)νG(ΠG(y))≕y+ξh(y,t),{\Xi }_{h}(y,t):= y+\chi \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t){\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\hspace{0.33em}=: \hspace{0.33em}y+{\xi }_{h}(y,t),where χ∈C∞(R)\chi \in {C}^{\infty }\left({\mathbb{R}})is a cutoff function such that 0≤χ≤10\le \chi \le 1, 1<∣χ′∣<31\lt | \chi ^{\prime} | \lt 3, χ(r)=1\chi \left(r)=1for ∣r∣≤1| r| \le 1, and χ(r)=0\chi \left(r)=0for ∣r∣>2| r| \gt 2. According to the definition of Ξh{\Xi }_{h}, we obtain Ξh(y,t)=zif∣dG(y)∣>2d,ΠG(Ξh(y,t))=ΠG(y)if∣dG(y)∣<d,dG(Ξh(y,t))=dG(y)+χdGdh(ΠG(y),t)if∣dG(y)∣<2d,\begin{array}{rcll}{\Xi }_{h}(y,t)& =& z& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \gt 2{\mathsf{d}},\\ {\Pi }_{{\mathscr{G}}}\left({\Xi }_{h}(y,t))& =& {\Pi }_{{\mathscr{G}}}(y)& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \lt {\mathsf{d}},\\ {d}_{{\mathscr{G}}}\left({\Xi }_{h}(y,t))& =& {d}_{{\mathscr{G}}}(y)+\chi \left(\frac{{d}_{{\mathscr{G}}}}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t)& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \lt 2{\mathsf{d}},\end{array}as well as Ξh−1(y,t)=y−h(ΠG(y),t)νG(ΠG(y))if∣dG(y)∣<d.{\Xi }_{h}^{-1}(y,t)=y-h\left({\Pi }_{{\mathscr{G}}}(y),t){\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\hspace{1.0em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}| {d}_{{\mathscr{G}}}(y)| \lt {\mathsf{d}}.By using the transform Ξh{\Xi }_{h}, we define the pull backs of V˜\widetilde{V}and P˜\widetilde{P}by u(y,t)≔V˜(Ξh(y,t),t),q(y,t)≔P˜(Ξh(y,t),t),y∈F,t>0,u(y,t):= \widetilde{V}\left({\Xi }_{h}(y,t),t),\hspace{1.0em}q(y,t):= \widetilde{P}\left({\Xi }_{h}(y,t),t),\hspace{1.0em}y\in {\mathscr{F}},\hspace{1.0em}t\gt 0,respectively. Then, we observe that (u,q,h)\left(u,q,h)satisfies the following system: (3.3)∂tu−μℒ(h)u+2ω(e3×u)+G(h)q=R(u,h),inF,G(h)⋅u=0,inF,(μ(G(h)u+[G(h)u]⊤)−qI)νΓ˜(h)=σHΓ˜(h)+ω22∣z′∣2+p0νΓ˜(h),onG,∂th−u⋅νG=−u⋅a(h)−1∣F∣(νG−a(h))⋅∫Ω˜(t)V˜dy,onG,u(0)=u0,inF,h(0)=h0,onG.\left\{\begin{array}{ll}{\partial }_{t}u-\mu {\mathcal{ {\mathcal L} }}\left(h)u+2\omega \left({e}_{3}\times u)+{\mathcal{G}}\left(h)q=R\left(u,h),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathscr{F}},\\ {\mathcal{G}}\left(h)\cdot u=0,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathscr{F}},\\ (\mu \left({\mathcal{G}}\left(h)u+{\left[{\mathcal{G}}\left(h)u]}^{\top })-qI){\nu }_{\widetilde{\Gamma }}\left(h)=\left(\sigma {{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)+\frac{{\omega }^{2}}{2}| z^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{\widetilde{\Gamma }}\left(h),& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}},\\ {\partial }_{t}h-u\cdot {\nu }_{{\mathscr{G}}}=-u\cdot a\left(h)-\frac{1}{| {\mathscr{F}}| }\left({\nu }_{{\mathscr{G}}}-a\left(h))\cdot \mathop{\displaystyle \int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}{\rm{d}}y,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}},\\ u\left(0)={u}_{0},& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathscr{F}},\\ h\left(0)={h}_{0},& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\mathscr{G}}.\end{array}\right.Here, ℒ(h){\mathcal{ {\mathcal L} }}\left(h), G(h){\mathcal{G}}\left(h), and HΓ˜(h){{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)are the transformed Laplacian, gradient, and doubled mean curvature, respectively, while the functions R(u,h)R\left(u,h)and a(h)a\left(h)are defined below. Furthermore, it holds DΞh=I+Dξh,(DΞh)−1=I−(I+Dξh)−1ξh≕I−[M1(h)]⊤,D{\Xi }_{h}=I+D{\xi }_{h},\hspace{1.0em}{\left(D{\Xi }_{h})}^{-1}=I-{\left(I+D{\xi }_{h})}^{-1}{\xi }_{h}\hspace{0.33em}=: \hspace{0.33em}I-{\left[{M}_{1}\left(h)]}^{\top },where Dξh(y,t)=1aχ′dG(y)dh(ΠG(y),t)νG(ΠG(y))⊗νG(ΠG(y))+χdG(y)dνG(y)⊗M0(ΠG(y))∇Gh(ΠG(y),t)−χdG(y)dh(ΠG(y),t)LG(ΠG(y))M0(ΠG(y))PG(ΠG(y))\begin{array}{rcl}D{\xi }_{h}(y,t)& =& \frac{1}{a}\chi ^{\prime} \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t){\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\displaystyle \otimes {\nu }_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\\ & & +\chi \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right){\nu }_{{\mathscr{G}}}(y)\displaystyle \otimes {M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){\nabla }_{{\mathscr{G}}}h\left({\Pi }_{{\mathscr{G}}}(y),t)\\ & & -\chi \left(\frac{{d}_{{\mathscr{G}}}(y)}{{\mathsf{d}}}\right)h\left({\Pi }_{{\mathscr{G}}}(y),t){L}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)){M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){{\mathcal{P}}}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y))\end{array}if ∣dG(y)∣<2d| {d}_{{\mathscr{G}}}(y)| \lt 2{\mathsf{d}}, while Dξh(y,t)=0D{\xi }_{h}(y,t)=0if ∣dG(y)∣>2d| {d}_{{\mathscr{G}}}(y)| \gt 2{\mathsf{d}}. In particular, if ∣dG(y)∣<d| {d}_{{\mathscr{G}}}(y)| \lt {\mathsf{d}}, we have Dξh(y,t)=νG(y)⊗M0(ΠG(y))∇Gh(ΠG(y),t)−h(ΠG(y),t)LG(ΠG(y))M0(ΠG(y))PG(ΠG(y)).D{\xi }_{h}(y,t)={\nu }_{{\mathscr{G}}}(y)\otimes {M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){\nabla }_{{\mathscr{G}}}h\left({\Pi }_{{\mathscr{G}}}(y),t)-h\left({\Pi }_{{\mathscr{G}}}(y),t){L}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)){M}_{0}\left({\Pi }_{{\mathscr{G}}}(y)){{\mathcal{P}}}_{{\mathscr{G}}}\left({\Pi }_{{\mathscr{G}}}(y)).On the basis of these representations, we find that (I+Dξh)\left(I+D{\xi }_{h})is boundedly invertible if ∣h∣L∞(G)<13mind,1∣LG∣L∞(G)and∣∇Gh∣L∞(G)<13.| h{| }_{{L}^{\infty }\left({\mathscr{G}})}\lt \frac{1}{3}\min \left({\mathsf{d}},\frac{1}{| {L}_{{\mathscr{G}}}{| }_{{L}^{\infty }\left({\mathscr{G}})}}\right)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}| {\nabla }_{{\mathscr{G}}}h{| }_{{L}^{\infty }\left({\mathscr{G}})}\lt \frac{1}{3}.Then we can write [∇P˜]∘Ξh=G(h)q=[DΞh−1∘Ξh]⊤∇q=[(DΞh)−1]⊤∇q=∇q−M1(h)∇q,[divV˜]∘Ξh=G(h)⋅u=((I−M1(h))∇⋅)u,\begin{array}{rcl}\left[\nabla \widetilde{P}]\circ {\Xi }_{h}& =& {\mathcal{G}}\left(h)q={\left[D{\Xi }_{h}^{-1}\circ {\Xi }_{h}]}^{\top }\nabla q={\left[{\left(D{\Xi }_{h})}^{-1}]}^{\top }\nabla q=\nabla q-{M}_{1}\left(h)\nabla q,\\ {[}{\rm{div}}\hspace{0.33em}\widetilde{V}]\circ {\Xi }_{h}& =& {\mathcal{G}}\left(h)\cdot u=\left(\left(I-{M}_{1}\left(h))\nabla \cdot )u,\end{array}and [ΔV˜]∘Ξh=ℒ(h)u=[(I−M1(h))∇]⋅[(I−M1(h))∇h]=Δu−[M1(h)+[M1(h)]⊤−M1(h)[M1(h)]⊤]:∇2u+[[(I−M1(h)):∇M1(h)]⋅∇]u≕Δu−M2(h):∇2u−M3(h)⋅∇u.\begin{array}{rcl}\left[\Delta \widetilde{V}]\circ {\Xi }_{h}& =& {\mathcal{ {\mathcal L} }}\left(h)u\\ & =& \left[\left(I-{M}_{1}\left(h))\nabla ]\cdot \left[\left(I-{M}_{1}\left(h))\nabla h]\\ & =& \Delta u-\left[{M}_{1}\left(h)+{\left[{M}_{1}\left(h)]}^{\top }-{M}_{1}\left(h){\left[{M}_{1}\left(h)]}^{\top }]:{\nabla }^{2}u+\left[\left[\left(I-{M}_{1}\left(h)):\nabla {M}_{1}\left(h)]\cdot \nabla ]u\\ & =: & \Delta u-{M}_{2}\left(h):{\nabla }^{2}u-{M}_{3}\left(h)\cdot \nabla u.\end{array}Here, we have used the notation A:B=∑i,j=13aijbij=tr(AB⊤)A:B=\mathop{\sum }\limits_{i,j=1}^{3}{a}_{ij}{b}_{ij}={\rm{tr}}\left(A{B}^{\top }). In our analysis, it is required to obtain another representation formula for [divV˜]∘Ξh\left[{\rm{div}}\hspace{0.33em}\widetilde{V}]\circ {\Xi }_{h}. To this end, we use the L2{L}^{2}inner product in Ω˜(t)\widetilde{\Omega }\left(t), which is denoted by (⋅,⋅)Ω˜(t){\left(\cdot ,\cdot )}_{\widetilde{\Omega }\left(t)}. For any test function φ˜∈Cc∞(Ω˜(t))\widetilde{\varphi }\in {C}_{c}^{\infty }\left(\widetilde{\Omega }\left(t)), we write φ(y)=φ˜(z)\varphi (y)=\widetilde{\varphi }\left(z). Here, Cc∞(D){C}_{c}^{\infty }\left(D)is the set of all C∞{C}^{\infty }-functions on R3{{\mathbb{R}}}^{3}, which have compact supports contained in D⊂R3D\subset {{\mathbb{R}}}^{3}. In addition, the Jacobian of the transform Ξh(y,t){\Xi }_{h}(y,t)is denoted by J=J(h){\mathsf{J}}={\mathsf{J}}\left(h). Recalling the expression of DξhD{\xi }_{h}, we shall write J(h)=1+J0(h){\mathsf{J}}\left(h)=1+{{\mathsf{J}}}_{0}\left(h)with some function J0(h){{\mathsf{J}}}_{0}\left(h)that vanishes at h=0h=0. Then, we see that (divV˜,φ˜)Ω˜(t)=−(V˜,∇φ˜)Ω˜(t)=−(Ju,(I−M1(h))∇φ)F=(div((I−[M1(h)]⊤)Ju),φ)F=(J−1div((I−[M1(h)]⊤)Ju),φ˜)Ω˜(t),\begin{array}{rcl}{\left({\rm{div}}\widetilde{V},\widetilde{\varphi })}_{\widetilde{\Omega }\left(t)}& =& -{\left(\widetilde{V},\nabla \widetilde{\varphi })}_{\widetilde{\Omega }\left(t)}\\ & =& -{\left({\mathsf{J}}u,\left(I-{M}_{1}\left(h))\nabla \varphi )}_{{\mathscr{F}}}\\ & =& {\left({\rm{div}}\left(\left(I-{\left[{M}_{1}\left(h)]}^{\top }){\mathsf{J}}u),\varphi )}_{{\mathscr{F}}}\\ & =& {\left({{\mathsf{J}}}^{-1}{\rm{div}}\left(\left(I-{\left[{M}_{1}\left(h)]}^{\top }){\mathsf{J}}u),\widetilde{\varphi })}_{\widetilde{\Omega }\left(t)},\end{array}where (⋅,⋅)F{\left(\cdot ,\cdot )}_{{\mathscr{F}}}is the L2{L}^{2}inner product in F{\mathscr{F}}. This gives [divV˜]∘Ξh=divu−M1(h):∇u=J−1(div((I−[M1(h)]⊤)Ju)),\left[{\rm{div}}\hspace{0.33em}\widetilde{V}]\circ {\Xi }_{h}={\rm{div}}\hspace{0.33em}u-{M}_{1}\left(h):\nabla u={{\mathsf{J}}}^{-1}({\rm{div}}\hspace{0.33em}\left(\left(I-{\left[{M}_{1}\left(h)]}^{\top }){\mathsf{J}}u)),i.e., the divergence free condition (3.2)2{\left(3.2)}_{2}is rewritten as follows: divu=−J0(h)divu+(1+J0(h))M1(h):∇u=div((1+J0(h))[M1(h)]⊤u).{\rm{div}}\hspace{0.33em}u=-{{\mathsf{J}}}_{0}\left(h){\rm{div}}\hspace{0.33em}u+\left(1+{{\mathsf{J}}}_{0}\left(h)){M}_{1}\left(h):\nabla u={\rm{div}}\hspace{0.33em}(\left(1+{{\mathsf{J}}}_{0}\left(h)){\left[{M}_{1}\left(h)]}^{\top }u).Next, we note that [∂tV˜]∘Ξh=∂tu−[(∇V˜)∘Ξh](∂tΞh)=∂tu−[[DΞh−1∘Ξh]⊤∇u](∂tΞh)=∂tu−∇u[(I+Dξh)−1∂tξh]≕∂tu−M4(h)∇u,\begin{array}{rcl}\left[{\partial }_{t}\widetilde{V}]\circ {\Xi }_{h}& =& {\partial }_{t}u-\left[\left(\nabla \widetilde{V})\circ {\Xi }_{h}]\left({\partial }_{t}{\Xi }_{h})\\ & =& {\partial }_{t}u-\left[{\left[D{\Xi }_{h}^{-1}\circ {\Xi }_{h}]}^{\top }\nabla u]\left({\partial }_{t}{\Xi }_{h})\\ & =& {\partial }_{t}u-\nabla u\left[{\left(I+D{\xi }_{h})}^{-1}{\partial }_{t}{\xi }_{h}]\\ & =: & {\partial }_{t}u-{M}_{4}\left(h)\nabla u,\end{array}which implies R(u,h)=−(u⋅G(h)u)+M4(h)∇u.R\left(u,h)=-\left(u\cdot {\mathcal{G}}\left(h)u)+{M}_{4}\left(h)\nabla u.Furthermore, it holds νΓ˜=b(h)(νG−a(h)),a(h)=M0(h)∇Gh,b(h)=11+∣a(h)∣2,M0(h)=(I−hLG)−1,{\nu }_{\widetilde{\Gamma }}=b\left(h)\left({\nu }_{{\mathscr{G}}}-a\left(h)),\hspace{1.0em}a\left(h)={M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h,\hspace{1.0em}b\left(h)=\frac{1}{\sqrt{1+| a\left(h){| }^{2}}},\hspace{1.0em}{M}_{0}\left(h)={\left(I-h{L}_{{\mathscr{G}}})}^{-1},and VΓ˜=(∂tΞh)⋅νΓ˜=∂th(νΓ˜⋅νG)=b(h)∂th.{V}_{\widetilde{\Gamma }}=\left({\partial }_{t}{\Xi }_{h})\cdot {\nu }_{\widetilde{\Gamma }}={\partial }_{t}h\left({\nu }_{\widetilde{\Gamma }}\cdot {\nu }_{{\mathscr{G}}})=b\left(h){\partial }_{t}h.Here, νG{\nu }_{{\mathscr{G}}}and a(h)a\left(h)are linearly independent. The term ∫Ω^(t)V˜(y,t)dy{\int }_{\widehat{\Omega }\left(t)}\widetilde{V}(y,t){\rm{d}}ycan be read as follows: ∫Ω˜(t)V˜dy=∫FuJ(h)dz=∫Fudz+∫FuJ0(h)dz.\mathop{\int }\limits_{\widetilde{\Omega }\left(t)}\widetilde{V}{\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}u{\mathsf{J}}\left(h){\rm{d}}z=\mathop{\int }\limits_{{\mathscr{F}}}u{\rm{d}}z+\mathop{\int }\limits_{{\mathscr{F}}}u{{\mathsf{J}}}_{0}\left(h){\rm{d}}z.The doubled mean curvature HΓ˜{{\mathscr{H}}}_{\widetilde{\Gamma }}is given by (3.4)HΓ˜(h)=b(h)(tr[M0(h)(LG+∇Ga(h))]−b2(h)(M0(h)a(h))⋅([∇Ga(h)]a(h))).{{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)=b\left(h)({\rm{tr}}\left[{M}_{0}\left(h)\left({L}_{{\mathscr{G}}}+{\nabla }_{{\mathscr{G}}}a\left(h))]-{b}^{2}\left(h)\left({M}_{0}\left(h)a\left(h))\cdot \left(\left[{\nabla }_{{\mathscr{G}}}a\left(h)]a\left(h))).Its linearization at h=0h=0is given by (3.5)HΓ˜′(0)=trLG2+ΔG=HG2−2KG+ΔG,{{\mathscr{H}}}_{\widetilde{\Gamma }}^{^{\prime} }\left(0)={\rm{tr}}{L}_{{\mathscr{G}}}^{2}+{\Delta }_{{\mathscr{G}}}={{\mathscr{H}}}_{{\mathscr{G}}}^{2}-2{{\mathscr{K}}}_{{\mathscr{G}}}+{\Delta }_{{\mathscr{G}}},where ΔG{\Delta }_{{\mathscr{G}}}is the Laplace-Beltrami operator on G{\mathscr{G}}. Here and in the following, for sufficiently smooth functions a,b∈C(G){\mathsf{a}},{\mathsf{b}}\in C\left({\mathscr{G}})and F(a):G→Rk{\mathsf{F}}\left({\mathsf{a}}):{\mathscr{G}}\to {{\mathbb{R}}}^{k}, we use the notation F′(a)b≔ddsF(a+sb)∣s=0,{\mathsf{F}}^{\prime} \left({\mathsf{a}}){\mathsf{b}}:= \frac{{\rm{d}}}{{\rm{d}}s}{\mathsf{F}}\left({\mathsf{a}}+s{\mathsf{b}}){| }_{s=0},which denotes the first variation of F(a){\mathsf{F}}\left({\mathsf{a}}). We refer to [22, Ch. 2] for the derivations of (3.4) and (3.5).We now decompose the stress boundary condition into tangential and normal parts. Multiplying (3.3)3{\left(3.3)}_{3}with νG/b{\nu }_{{\mathscr{G}}}\hspace{0.1em}\text{/}\hspace{0.1em}b, we obtain q+σHΓ˜(h)+ω22∣z′∣2+p0=(μ(G(h)u+[G(h)u]⊤)(νG−a(h)))⋅νGq+\sigma {{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)+\frac{{\omega }^{2}}{2}| z^{\prime} {| }^{2}+{p}_{0}=(\mu \left({\mathcal{G}}\left(h)u+{\left[{\mathcal{G}}\left(h)u]}^{\top })\left({\nu }_{{\mathscr{G}}}-a\left(h)))\cdot {\nu }_{{\mathscr{G}}}for the normal part of (3.3)3{\left(3.3)}_{3}, while (3.6)PG(μ(G(h)u+[G(h)u]⊤)(νG−a(h)))=0{{\mathcal{P}}}_{{\mathscr{G}}}(\mu \left({\mathcal{G}}\left(h)u+{\left[{\mathcal{G}}\left(h)u]}^{\top })\left({\nu }_{{\mathscr{G}}}-a\left(h)))=0for the tangential part of (3.3)3{\left(3.3)}_{3}. It should be emphasized that (3.6) neither contains the pressure nor the curvature. Finally, we have ∣z′∣2−∣y′∣2−∂∂νG∣y′∣2h=h2((νG(1))2+(νG(2))2).| z^{\prime} {| }^{2}-| y^{\prime} {| }^{2}-\left(\frac{\partial }{\partial {\nu }_{{\mathscr{G}}}}| y^{\prime} {| }^{2}\right)h={h}^{2}({\left({\nu }_{{\mathscr{G}}}^{\left(1)})}^{2}+{\left({\nu }_{{\mathscr{G}}}^{\left(2)})}^{2}).Consequently, from the aforementioned discussion, problem (3.2) can be rewritten as follows: (3.7)∂tu−μΔu+2ω(e3×u)+∇q=Fu(u,q,h),in F,divu=Gd(u,h)=divGdiv(u,h),in F,PG(μ(∇u+[∇u]⊤)νG)=Guτ(u,h),on G,μ(∇u+[∇u]⊤)νG⋅νG−q+ℬGh=Guv(u,h)+G0(h),on G,∂th−u⋅νG+1∣F∣νG⋅∫Fudz=Fh(u,h)+F(u,h),on G,u(0)=u0,in F,h(0)=h0,on G.\left\{\begin{array}{ll}{\partial }_{t}u-\mu \Delta u+2\omega \left({e}_{3}\times u)+\nabla q={F}_{u}\left(u,q,h),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}u={G}_{d}\left(u,h)={\rm{div}}\hspace{0.33em}{G}_{{\rm{div}}}\left(u,h),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}(\mu \left(\nabla u+{\left[\nabla u]}^{\top }){\nu }_{{\mathscr{G}}})={G}_{u\tau }\left(u,h),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \mu \left(\nabla u+{\left[\nabla u]}^{\top }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-q+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h={G}_{uv}\left(u,h)+{G}_{0}\left(h),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}h-u\cdot {\nu }_{{\mathscr{G}}}+\frac{1}{| {\mathscr{F}}| }{\nu }_{{\mathscr{G}}}\cdot \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}u{\rm{d}}z={F}_{h}\left(u,h)+F\left(u,h),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ u\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ h\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Here, the right-hand members of (3.7) can be written as follows: Fu(u,q,h)=M4(h)∇u−u⋅(I−M1(h))∇u−μ(M2(h):∇2u+M3(h)⋅∇u)+M1(h)∇q,Gd(u,h)=−J0(h)divu+(1+J0(h))M1(h):∇u,Gdiv(u,h)=(1+J0(h))⊤M1(h)u,Guτ(u,h)=PG(μ(M1(h)∇u+[M1(h)∇u]⊤)(νG−M0(h)∇Gh))+PG((∇u+[∇u]⊤)M0(h)∇Gh),Guv(u,h)=−(∇u+[∇u]⊤)M0(h)∇Gh⋅νG+(M1(h)∇u+[M1(h)∇u]⊤)(νG−M0(h)∇Gh)⋅νGG0(h)=σ(HΓ˜(h)−HΓ˜′(0)h)+ω22{h2((νG(1))2+(νG(2))2)},Fh(u,h)=−(M0(h)∇Gh)⋅u,F(u,h)=1∣F∣a(h)⋅∫Fu(1+J0)dz−1∣F∣νG⋅∫FuJ0(h)dz.\begin{array}{rcl}{F}_{u}\left(u,q,h)& =& {M}_{4}\left(h)\nabla u-u\cdot \left(I-{M}_{1}\left(h))\nabla u-\mu ({M}_{2}\left(h):{\nabla }^{2}u+{M}_{3}\left(h)\cdot \nabla u)+{M}_{1}\left(h)\nabla q,\\ {G}_{d}\left(u,h)& =& -{{\mathsf{J}}}_{0}\left(h){\rm{div}}\hspace{0.33em}u+\left(1+{{\mathsf{J}}}_{0}\left(h)){M}_{1}\left(h):\nabla u,\\ {G}_{{\rm{div}}}\left(u,h)& =& {\left(1+{{\mathsf{J}}}_{0}\left(h))}^{\top }{M}_{1}\left(h)u,\\ {G}_{u\tau }\left(u,h)& =& {{\mathcal{P}}}_{{\mathscr{G}}}(\mu \left({M}_{1}\left(h)\nabla u+{\left[{M}_{1}\left(h)\nabla u]}^{\top })\left({\nu }_{{\mathscr{G}}}-{M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h))+{{\mathcal{P}}}_{{\mathscr{G}}}(\left(\nabla u+{\left[\nabla u]}^{\top }){M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h),\\ {G}_{uv}\left(u,h)& =& -\left(\nabla u+{\left[\nabla u]}^{\top }){M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h\cdot {\nu }_{{\mathscr{G}}}+\left({M}_{1}\left(h)\nabla u+{\left[{M}_{1}\left(h)\nabla u]}^{\top })\left({\nu }_{{\mathscr{G}}}-{M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h)\cdot {\nu }_{{\mathscr{G}}}\\ {G}_{0}\left(h)& =& \sigma \left({{\mathscr{H}}}_{\widetilde{\Gamma }}\left(h)-{{\mathscr{H}}}_{\widetilde{\Gamma }}^{^{\prime} }\left(0)h)+\frac{{\omega }^{2}}{2}\{{h}^{2}({\left({\nu }_{{\mathscr{G}}}^{\left(1)})}^{2}+{\left({\nu }_{{\mathscr{G}}}^{\left(2)})}^{2})\},\\ {F}_{h}\left(u,h)& =& -\left({M}_{0}\left(h){\nabla }_{{\mathscr{G}}}h)\cdot u,\\ F\left(u,h)& =& \frac{1}{| {\mathscr{F}}| }a\left(h)\cdot \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}u\left(1+{{\mathsf{J}}}_{0}){\rm{d}}z-\frac{1}{| {\mathscr{F}}| }{\nu }_{{\mathscr{G}}}\cdot \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}u{{\mathsf{J}}}_{0}\left(h){\rm{d}}z.\end{array}Notice that the right-hand members of (3.7) vanish at (u,q,h)=(0,0,0)\left(u,q,h)=\left(0,0,0).4Maximal regularityTo simplify the notation, in the following, we write D(u)≔2−1(∇u+[∇u]⊤)D\left(u):= {2}^{-1}\left(\nabla u+{\left[\nabla u]}^{\top })for every vector field uu. The principal part of the linearized problem reads as follows: (4.1)∂tu−μΔu+2ω(e3×u)+∇q=fu,in F,divu=gd,in F,PG(2μD(u)νG)=guτ,on G,2μD(u)νG⋅νG−q+ℬGh=guv,on G,∂th−(P0Gu)⋅νG=fh,on G,u(0)=u0,in F,h(0)=h0,on G,\left\{\begin{array}{ll}{\partial }_{t}u-\mu \Delta u+2\omega \left({e}_{3}\times u)+\nabla q={f}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}u={g}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(u){\nu }_{{\mathscr{G}}})={g}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(u){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-q+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}h={g}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}h-\left({P}_{0}^{{\mathscr{G}}}u)\cdot {\nu }_{{\mathscr{G}}}={f}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ u\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ h\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where we have set P0Gu=u−1∣F∣∫Fu(y,t)dy.{P}_{0}^{{\mathscr{G}}}u=u-\frac{1}{| {\mathscr{F}}| }\mathop{\int }\limits_{{\mathscr{F}}}u(y,t){\rm{d}}y.We define the function spaces Eδ(J;F)≔E1,δ(J;F)×E2,δ(J;F)×E3,δ(J;G)×E4,δ(J;G),E1,δ(J;F)≔Hδ1,p(J;Lq(F)3)∩Lδp(J;H2,q(F)3),E2,δ(J;F)≔Lδp(J;H˙1,q(F)),E3,δ(J;G)≔Fp,q,δ1/2−1/(2q)(J;Lq(G))∩Lδp(J;Bq,q1−1/q(G)),E4,δ(J;G)≔Fp,q,δ2−1/q(J;Lq(G))∩Hδ1,p(J;Bq,q2−1/q(G))∩Lδp(J;Bq,q3−1/q(G)),Fδ(J;F)≔F1,δ(J;F)×F2,δ(J;F)×F3,δ(J;F)2×F4,δ(J;G),F1,δ(J;F)≔Lδp(J;Lq(F)3),F2,δ(J;F)≔Hδ1,p(J;H˙−1,q(F))∩Lδp(J;DIq(F)),F3,δ(J;G)≔Fp,q,δ1/2−1/(2q)(J;Lq(G))∩Lδp(J;Bq,q1−1/q(G)),F4,δ(J;G)≔Fp,q,δ1−1/(2q)(J;Lq(G))∩Lδp(J;Bq,q2−1/q(G))\begin{array}{rcl}{{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {{\mathbb{E}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{E}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})\times {{\mathbb{E}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}}),\\ {{\mathbb{E}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {H}_{\delta }^{1,p}\left(J;\hspace{0.33em}{L}^{q}{\left({\mathscr{F}})}^{3})\cap {L}_{\delta }^{p}\left(J;{H}^{2,q}{\left({\mathscr{F}})}^{3}),\\ {{\mathbb{E}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {L}_{\delta }^{p}\left(J;\hspace{0.33em}{\dot{H}}^{1,q}\left({\mathscr{F}})),\\ {{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{1\hspace{0.1em}\text{/}2-1\text{/}\hspace{0.1em}\left(2q)}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{1-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})),\\ {{\mathbb{E}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {H}_{\delta }^{1,p}\left(J;\hspace{0.33em}{B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})),\\ {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {{\mathbb{F}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{F}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\times {{\mathbb{F}}}_{3,\delta }{\left(J;{\mathscr{F}})}^{2}\times {{\mathbb{F}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}}),\\ {{\mathbb{F}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {L}_{\delta }^{p}\left(J;\hspace{0.33em}{L}^{q}{\left({\mathscr{F}})}^{3}),\\ {{\mathbb{F}}}_{2,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})& := & {H}_{\delta }^{1,p}\left(J;\hspace{0.33em}{\dot{H}}^{-1,q}\left({\mathscr{F}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{{\rm{DI}}}_{q}\left({\mathscr{F}})),\\ {{\mathbb{F}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{1\hspace{0.1em}\text{/}2-1\text{/}\hspace{0.1em}\left(2q)}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{1-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})),\\ {{\mathbb{F}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})& := & {F}_{p,q,\delta }^{1-1\hspace{0.1em}\text{/}\hspace{0.1em}\left(2q)}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))\cap {L}_{\delta }^{p}\left(J;\hspace{0.33em}{B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))\end{array}for J⊂R+J\subset {{\mathbb{R}}}_{+}. The main theorem of this section states that problem (4.1) has maximal regularity.Theorem 4.1Let 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. There exists a constant β0>0{\beta }_{0}\gt 0such that for all β≥β0\beta \ge {\beta }_{0}, problem (4.1) has a unique solution (u,q,TrG[q],h)∈eβtEδ(R+;F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {e}^{\beta t}{{\mathbb{E}}}_{\delta }\left({{\mathbb{R}}}_{+};\hspace{0.33em}{\mathscr{F}})if and only if(a)(u0,h0)∈Bq,p2(δ−1/p)(F)3×Bq,p2+δ−1/p−1/q(G)\left({u}_{0},{h}_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}});(b)(fu,gd,guτ,guv,fh)∈eβtFδ(R+;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {e}^{\beta t}{{\mathbb{F}}}_{\delta }\left({{\mathbb{R}}}_{+};\hspace{0.33em}{\mathscr{F}});(c)gd∣t=0=divu0{g}_{d}{| }_{t=0}={\rm{div}}\hspace{0.33em}{u}_{0};(d)guτ∣t=0=PG(2μD(u)νG){g}_{u\tau }{| }_{t=0}={{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(u){\nu }_{{\mathscr{G}}})if 1/p+1/(2q)<δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\lt \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2.Furthermore, the solution (u,q,TrG[q],h)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)enjoys the estimate∣e−βt(u,q,TrG[q],h)∣Eδ(R+;F)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣η0∣Bq,p2+δ−1/p−1/q(G)+∣e−βt(fu,gd,guτ,guv,fh)∣Fδ(R+;F))| {e}^{-\beta t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left({{\mathbb{R}}}_{+};{\mathscr{F}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {\eta }_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{-\beta t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left({{\mathbb{R}}}_{+};{\mathscr{F}})})with some constant C independent of ω\omega , β0{\beta }_{0}, β\beta , and TT.To prove Theorem 4.1, we consider the corresponding resolvent problem: (4.2)λu^−μΔu^+2ω(e3×u^)+∇q^=f^u,in F,divu^=g^d,in F,PG(2μD(u^)νG)=g^uτ,on G,2μD(u^)νG⋅νG−q^+ℬGh^=g^uv,on G,λh^−(P0Gu^)⋅νG=f^h,on G,\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q}={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}={\widehat{g}}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})={\widehat{g}}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h}={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.which can be obtained by the Laplace transform, with respect to time tt, applied to the system (4.1). Here, λ\lambda is the resolvent parameter varying in Σε,λ0≔{λ∈C:∣argλ∣≤π−ε,∣λ∣≥λ0}{\Sigma }_{\varepsilon ,{\lambda }_{0}}:= \left\{\lambda \in {\mathbb{C}}\hspace{0.33em}:\hspace{0.33em}| \arg \lambda | \le \pi -\varepsilon ,| \lambda | \ge {\lambda }_{0}\right\}for ε∈(0,π/2)\varepsilon \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2)and λ0>0{\lambda }_{0}\gt 0. By using the result obtained by Shibata [27, Thm. 4.8], we show the existence of ℛ{\mathcal{ {\mathcal R} }}-bounded solution operator families for (4.2). This section is mainly devoted to the proof of the following lemma.Lemma 4.2Assume 1<q<∞1\lt q\lt \infty and 0<ε<π/20\lt \varepsilon \lt \pi \hspace{0.1em}\text{/}\hspace{0.1em}2. LetXq≔(f^u,g^d,g^uτ,g^uv,f^h):f^u∈Lq(F)3,g^d∈DIq(F),g^uτ∈H1,q(F)2,g^uv∈H1,q(F),f^h∈Bq,q2−1/q(G),Xq≔F=(F1,…,F9):F1,F4∈Lq(F)3,F2∈Lq(F),F3∈H1,q(F),F5,F7∈Lq(F)2,F6,F8∈H1,q(F)2,F9∈Bq,q2−1/q(G)..\begin{array}{rcl}{X}_{q}& := & \left\{({\widehat{f}}_{u},{\widehat{g}}_{d},{\widehat{g}}_{u\tau },{\widehat{g}}_{uv},{\widehat{f}}_{h})\hspace{0.33em}:\hspace{0.33em}\begin{array}{c}{\widehat{f}}_{u}\in {L}^{q}{\left({\mathscr{F}})}^{3},{\widehat{g}}_{d}\in {{\rm{DI}}}_{q}\left({\mathscr{F}}),{\widehat{g}}_{u\tau }\in {H}^{1,q}{\left({\mathscr{F}})}^{2},\\ {\widehat{g}}_{uv}\in {H}^{1,q}\left({\mathscr{F}}),{\widehat{f}}_{h}\in {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})\end{array}\right\},\\ {{\mathcal{X}}}_{q}& := & \left\{F=\left({F}_{1},\ldots ,{F}_{9})\hspace{0.33em}:\hspace{0.33em}\begin{array}{c}{F}_{1},{F}_{4}\in {L}^{q}{\left({\mathscr{F}})}^{3},{F}_{2}\in {L}^{q}\left({\mathscr{F}}),{F}_{3}\in {H}^{1,q}\left({\mathscr{F}}),\\ {F}_{5},{F}_{7}\in {L}^{q}{\left({\mathscr{F}})}^{2},{F}_{6},{F}_{8}\in {H}^{1,q}{\left({\mathscr{F}})}^{2},{F}_{9}\in {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}).\end{array}\right\}.\end{array}Then there exists a constant λ1≥C0max(ω2,1){\lambda }_{1}\ge {C}_{0}\max \left({\omega }^{2},1)with some constant C0{C}_{0}and families of operators Uω(λ){{\mathcal{U}}}_{\omega }\left(\lambda ), Pω(λ){{\mathcal{P}}}_{\omega }\left(\lambda ), and ℋω(λ){{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )withUω(λ)∈Hol(Σε,λ1;ℒ(Xq,H2,q(F)3)),Qω(λ)∈Hol(Σε,λ1;ℒ(Xq,H1,q(F))),ℋω(λ)∈Hol(Σε,λ1;ℒ(Xq,Bq,q3−1/q(G))),\begin{array}{rcl}{{\mathcal{U}}}_{\omega }\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{1}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{2,q}{\left({\mathscr{F}})}^{3})),\\ {{\mathcal{Q}}}_{\omega }\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{1}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{1,q}\left({\mathscr{F}}))),\\ {{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{1}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}))),\end{array}such that (u^,q^,η^)≔(Uω(λ),Pω(λ),ℋω(λ))FλG\left(\widehat{u},\widehat{q},\widehat{\eta }):= \left({{\mathcal{U}}}_{\omega }\left(\lambda ),{{\mathcal{P}}}_{\omega }\left(\lambda ),{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )){F}_{\lambda }Gis the unique solution to (4.2), where G and Fλ{F}_{\lambda }are given byG≔(f^u,g^d,G(g^d),g^uτ,g^uv,f^h),FλG≔(f^u,λ1/2g^d,g^d,λG(g^d),λ1/2g^uτ,g^uτ,λ1/2g^uv,g^uv,f^h).\begin{array}{rcl}G& := & ({\widehat{f}}_{u},{\widehat{g}}_{d},{\mathsf{G}}\left({\widehat{g}}_{d}),{\widehat{g}}_{u\tau },{\widehat{g}}_{uv},{\widehat{f}}_{h}),\\ {F}_{\lambda }G& := & ({\widehat{f}}_{u},{\lambda }^{1\text{/}2}{\widehat{g}}_{d},{\widehat{g}}_{d},\lambda {\mathsf{G}}\left({\widehat{g}}_{d}),{\lambda }^{1\text{/}2}{\widehat{g}}_{u\tau },{\widehat{g}}_{u\tau },{\lambda }^{1\text{/}2}{\widehat{g}}_{uv},{\widehat{g}}_{uv},{\widehat{f}}_{h}).\end{array}Besides, it holds(4.3)ℛXq→H2−j,q(F)3({(τ∂τ)ℓ(λj/2Uω(λ)):λ∈Σε,λ1})≤c,ℛXq→Lq(F)3({(τ∂τ)ℓ∇Qω(λ):λ∈Σε,λ1})≤c,ℛXq→Bq,q3−1/q−k(G)({(τ∂τ)ℓ(λkℋω(λ)):λ∈Σε,λ1})≤c\begin{array}{rcl}{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {H}^{2-j,q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{j\text{/}2}{{\mathcal{U}}}_{\omega }\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {L}^{q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\nabla {{\mathcal{Q}}}_{\omega }\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {B}_{q,q}^{3-1\text{/}q-k}\left({\mathscr{G}})}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{k}{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})& \le & c\end{array}for ℓ=0,1\ell =0,1, j=0,1,2j=0,1,2, k=0,1k=0,1, and τ≔Imλ\tau := {\rm{Im}}\hspace{0.33em}\lambda , where c is independent of ω\omega .ProofAccording to Shibata [27, Thm. 4.8], we know the existence of ℛ{\mathcal{ {\mathcal R} }}-bounded solution operator for the following problem: (4.4)λu^−μΔu^+∇q^=f^u,in F,divu^=g^d,in F,PG(2μD(u^)νG)=g^uτ,on G,2μD(u^)νG⋅νG−q^−σH′(0)h^=g^uv,on G,λh^−(P0Gu^)⋅νG=f^h,on G.\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+\nabla \widehat{q}={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}={\widehat{g}}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})={\widehat{g}}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}-\sigma {\mathscr{H}}^{\prime} \left(0)\widehat{h}={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.More precisely, he prove that there exist a constant λ0{\lambda }_{0}and families of operators U0(λ){{\mathcal{U}}}_{0}\left(\lambda ), P0(λ){{\mathcal{P}}}_{0}\left(\lambda ), and ℋ0(λ){{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )with U0(λ)∈Hol(Σε,λ0;ℒ(Xq,H2,q(F)3)),Q0(λ)∈Hol(Σε,λ0;ℒ(Xq,H1,q(F))),ℋ0(λ)∈Hol(Σε,λ0;ℒ(Xq,Bq,q3−1/q(G)))\begin{array}{rcl}{{\mathcal{U}}}_{0}\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{0}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{2,q}{\left({\mathscr{F}})}^{3})),\\ {{\mathcal{Q}}}_{0}\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{0}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{H}^{1,q}\left({\mathscr{F}}))),\\ {{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )& \in & {\rm{Hol}}\left({\Sigma }_{\varepsilon ,{\lambda }_{0}};\hspace{0.33em}{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q},{B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})))\end{array}such that for every λ∈Σε,λ0\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}and G∈XqG\in {X}_{q}, the triplet (u^,q^,h^)=(U0(λ),Q0(λ),ℋ0(λ))FλG\left(\widehat{u},\widehat{q},\widehat{h})=\left({{\mathcal{U}}}_{0}\left(\lambda ),{{\mathcal{Q}}}_{0}\left(\lambda ),{{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )){F}_{\lambda }Gis a unique solution to problem (4.4) satisfying (4.5)ℛXq→H2−j,q(F)3({(τ∂τ)ℓ(λj/2U0(λ)):λ∈Σε,λ0})≤c0,ℛXq→Lq(F)3({(τ∂τ)ℓ∇Q0(λ):λ∈Σε,λ0})≤c0,ℛXq→Bq,q3−1/q−k(G)({(τ∂τ)ℓ(λkℋ0(λ)):λ∈Σε,λ0})≤c0\begin{array}{rcl}{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {H}^{2-j,q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{j\text{/}2}{{\mathcal{U}}}_{0}\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}\})& \le & {c}_{0},\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {L}^{q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\nabla {{\mathcal{Q}}}_{0}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}\})& \le & {c}_{0},\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {B}_{q,q}^{3-1\text{/}q-k}\left({\mathscr{G}})}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{k}{{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda ))\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{0}}\})& \le & {c}_{0}\end{array}for ℓ=0,1\ell =0,1, j=0,1,2j=0,1,2, k=0,1k=0,1, and τ≔Imλ\tau := {\rm{Im}}\hspace{0.33em}\lambda . Here, λ0{\lambda }_{0}and c0{c}_{0}are independent of ω\omega . Then we see that the solution (u^,q^,h^)\left(\widehat{u},\widehat{q},\widehat{h})of (4.4) satisfies λu^−μΔu^+2ω(e3×u^)+∇q^=f^u+2ω(e3×u^),in F,divu^=g^d,in F,PG(2μD(u^)νG)=g^uτ,on G,2μD(u^)νG⋅νG−q^+ℬGh^=g^uv+ω2(y′⋅νG)h^,on G,λh^−(P0Gu^)⋅νG=f^h,on G.\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q}={\widehat{f}}_{u}+2\omega \left({e}_{3}\times \widehat{u}),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}={\widehat{g}}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})={\widehat{g}}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h}={\widehat{g}}_{uv}+{\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}})\widehat{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Now we define R1(λ)G≔−2ω(e3×U0(λ)FλG),R1(λ)F≔−2ω(e3×U0(λ)F),R2(λ)G≔−ω2(y′⋅νG)ℋ0(λ)FλG,R2(λ)F≔−ω2(y′⋅νG)ℋ0(λ)F\begin{array}{rcl}{R}_{1}\left(\lambda )G& := & -2\omega \left({e}_{3}\times {{\mathcal{U}}}_{0}\left(\lambda ){F}_{\lambda }G),\hspace{1.0em}{{\mathscr{R}}}_{1}\left(\lambda )F:= -2\omega \left({e}_{3}\times {{\mathcal{U}}}_{0}\left(\lambda )F),\\ {R}_{2}\left(\lambda )G& := & -{\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}}){{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda ){F}_{\lambda }G,\hspace{1.0em}{{\mathscr{R}}}_{2}\left(\lambda )F:= -{\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}}){{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda )F\end{array}for G∈XqG\in {X}_{q}and F∈XqF\in {{\mathcal{X}}}_{q}. Setting R(λ)≔(R1(λ),0,0,R2(λ),0)R\left(\lambda ):= \left({R}_{1}\left(\lambda ),0,0,{R}_{2}\left(\lambda ),0)and R(λ)≔(R1(λ),0,0,R2(λ),0){\mathscr{R}}\left(\lambda ):= \left({{\mathscr{R}}}_{1}\left(\lambda ),0,0,{{\mathscr{R}}}_{2}\left(\lambda ),0), we have the relation (4.6)R(λ)=R(λ)Fλ,R\left(\lambda )={\mathscr{R}}\left(\lambda ){F}_{\lambda },which maps from Xq{X}_{q}to Xq{{\mathcal{X}}}_{q}. Since it holds ∣2ω(e3×u^)∣Lq(F)≤2ω∣u^∣Lq(F),∣ω2(y′⋅νG)h^∣H1,q(F)≤Cω2∣y′⋅νG∣H1,∞(F)∣h^∣H1,q(F)≤Cq,Gω2∣h^∣Bq,q2−1/q(G)\begin{array}{rcl}| 2\omega \left({e}_{3}\times \widehat{u}){| }_{{L}^{q}\left({\mathscr{F}})}& \le & 2\omega | \widehat{u}{| }_{{L}^{q}\left({\mathscr{F}})},\\ | {\omega }^{2}(y^{\prime} \cdot {\nu }_{{\mathscr{G}}})\widehat{h}{| }_{{H}^{1,q}\left({\mathscr{F}})}& \le & C{\omega }^{2}| y^{\prime} \cdot {\nu }_{{\mathscr{G}}}{| }_{{H}^{1,\infty }\left({\mathscr{F}})}| \widehat{h}{| }_{{H}^{1,q}\left({\mathscr{F}})}\le {C}_{q,{\mathscr{G}}}{\omega }^{2}| \widehat{h}{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}\end{array}for (u^,h^)∈H2,q(F)3×Bq,q3−1/q(G)\left(\widehat{u},\widehat{h})\in {H}^{2,q}{\left({\mathscr{F}})}^{3}\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), q∈(1,∞)q\in \left(1,\infty ), it follows from Proposition 2.1 and (4.5) that ℛXq({(τ∂τ)ℓFλR(λ):λ∈Σε,λ1})≤c0(2ωλ1−1+Cq,Gω2λ1−1){{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }{F}_{\lambda }{\mathscr{R}}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})\le {c}_{0}\left(2\omega {\lambda }_{1}^{-1}+{C}_{q,{\mathscr{G}}}{\omega }^{2}{\lambda }_{1}^{-1})for any λ1≥λ0{\lambda }_{1}\ge {\lambda }_{0}. We shall choose λ1{\lambda }_{1}large enough such that λ1≥4(Cq,G+1)c0max(1,ω2)≕C0max(1,ω2).{\lambda }_{1}\ge 4\left({C}_{q,{\mathscr{G}}}+1){c}_{0}\max \left(1,{\omega }^{2})\hspace{0.33em}=: \hspace{0.33em}{C}_{0}\max \left(1,{\omega }^{2}).Then, we have (4.7)ℛXq({(τ∂τ)ℓFλR(λ):λ∈Σε,λ1})≤12.{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }{F}_{\lambda }{\mathscr{R}}\left(\lambda )\hspace{0.33em}:\hspace{0.33em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}\})\le \frac{1}{2}.Hence, from (4.6) and (4.7), we obtain ∣FλR(λ)G∣Xq≤12∣FλG∣Xq.| {F}_{\lambda }R\left(\lambda )G{| }_{{{\mathcal{X}}}_{q}}\le \frac{1}{2}| {F}_{\lambda }G{| }_{{{\mathcal{X}}}_{q}}.Namely, we have FλR(λ)Fλ−1∈ℒ(Xq){F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1}\in {\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q})with ∣FλR(λ)Fλ−1∣ℒ(Xq)≤1/2| {F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1}{| }_{{\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q})}\le 1\hspace{0.1em}\text{/}\hspace{0.1em}2. Then, the Neumann series argument implies the existence of the inverse (I−FλR(λ)Fλ−1)−1{\left(I-{F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1})}^{-1}of FλR(λ)Fλ−1{F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1}in ℒ(Xq){\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q}). Hence, the operator Fλ−1(I−FλR(λ)Fλ−1)−1Fλ=(I−R(λ))−1{F}_{\lambda }^{-1}{\left(I-{F}_{\lambda }R\left(\lambda ){F}_{\lambda }^{-1})}^{-1}{F}_{\lambda }={\left(I-R\left(\lambda ))}^{-1}exists in ℒ(Xq){\mathcal{ {\mathcal L} }}\left({{\mathcal{X}}}_{q})for each λ∈Σε,λ1\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}. Setting Uω(λ)≔U0(λ)(I−R(λ))−1,Qω(λ)≔Q0(λ)(I−R(λ))−1,ℋω(λ)≔ℋ0(λ)(I−R(λ))−1,{{\mathcal{U}}}_{\omega }\left(\lambda ):= {{\mathcal{U}}}_{0}\left(\lambda ){\left(I-R\left(\lambda ))}^{-1},\hspace{1.0em}{{\mathcal{Q}}}_{\omega }\left(\lambda ):= {{\mathcal{Q}}}_{0}\left(\lambda ){\left(I-R\left(\lambda ))}^{-1},\hspace{1.0em}{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ):= {{\mathcal{ {\mathcal H} }}}_{0}\left(\lambda ){\left(I-R\left(\lambda ))}^{-1},we see that (Uω(λ),Qω(λ),ℋω(λ))FλG\left({{\mathcal{U}}}_{\omega }\left(\lambda ),{{\mathcal{Q}}}_{\omega }\left(\lambda ),{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )){F}_{\lambda }Gsolves (4.2) such that the estimates (4.3) are valid with c=4c0c=4{c}_{0}. Finally, the uniqueness of solutions to (4.2) follows from the duality argument. Suppose that u^∈H2,q(F)3\widehat{u}\in {H}^{2,q}{\left({\mathscr{F}})}^{3}, q^∈H1,q(F)\widehat{q}\in {H}^{1,q}\left({\mathscr{F}}), and h^∈Bq,q3−1/q(G)\widehat{h}\in {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfy problem (4.2) with (f^u,g^d,g^uτ,g^uv,f^h)({\widehat{f}}_{u},{\widehat{g}}_{d},{\widehat{g}}_{u\tau },{\widehat{g}}_{uv},{\widehat{f}}_{h})vanishing. For any λ∈Σε,λ1\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}, Φ∈Cc∞(F)3\Phi \in {C}_{c}^{\infty }{\left({\mathscr{F}})}^{3}, we consider that (4.8)λ¯v^−μΔv^+2(−ω)(e3×v^)+∇p^=Φ,in F,divv^=0,in F,PG(2μD(v^)νG)=0,on G,2μD(v^)νG⋅νG−p^+ℬGθ^=0,on G,λ¯θ^−(P0Gv^)⋅νG=0,on G,\left\{\begin{array}{ll}\overline{\lambda }\widehat{v}-\mu \Delta \widehat{v}+2\left(-\omega )\left({e}_{3}\times \widehat{v})+\nabla \widehat{p}=\Phi ,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{v}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{v}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{v}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{p}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \overline{\lambda }\widehat{\theta }-\left({P}_{0}^{{\mathscr{G}}}\widehat{v})\cdot {\nu }_{{\mathscr{G}}}=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where λ¯\overline{\lambda }stands for the complex conjugate of λ\lambda . Then, according to the aforementioned discussion, there exists a solution (v^,p^,θ^)∈H2,q(F)3×H1,q(F)×Bq,q2−1/q(G)\left(\widehat{v},\widehat{p},\widehat{\theta })\in {H}^{2,q}{\left({\mathscr{F}})}^{3}\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})of (4.8) with 1<q<∞1\lt q\lt \infty . Hence, the divergence theorem gives (u^,Φ)F=(u^,λ¯v^)F−(u^,−μΔv^+∇p^)F+(u^,2(−ω)(e3×v^))F=(u^,λ¯v^)F−(u^,(2μD(v^)−p^I)νG)G+2μ(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v^)F−(u^⋅νG,−ℬGθ^)G+2μ(D(u^),D(v^))F+(2ω(e3×u^),v^)F.\begin{array}{rcl}{\left(\widehat{u},\Phi )}_{{\mathscr{F}}}& =& {\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}-{\left(\widehat{u},-\mu \Delta \widehat{v}+\nabla \widehat{p})}_{{\mathscr{F}}}+{\left(\widehat{u},2\left(-\omega )\left({e}_{3}\times \widehat{v}))}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}-{\left(\widehat{u},\left(2\mu D\left(\widehat{v})-\widehat{p}I){\nu }_{{\mathscr{G}}})}_{{\mathscr{G}}}+2\mu {\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}-{\left(\widehat{u}\cdot {\nu }_{{\mathscr{G}}},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+2\mu {\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}.\end{array}Here, we have used the identity (e3×u^,v^)F=−(u^,e3×v^)F{\left({e}_{3}\times \widehat{u},\widehat{v})}_{{\mathscr{F}}}=-{\left(\widehat{u},{e}_{3}\times \widehat{v})}_{{\mathscr{F}}}. By (4.8)5{\left(4.8)}_{5}, it holds (u^⋅νG,−ℬGθ^)G=(λh^,−ℬGθ^)G+1∣F∣∫Fv^dy,−ℬGθ^G=(λh^,−ℬGθ^)G,{\left(\widehat{u}\cdot {\nu }_{{\mathscr{G}}},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}={\left(\lambda \widehat{h},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+{\left(\frac{1}{| {\mathscr{F}}| }\mathop{\int }\limits_{{\mathscr{F}}}\widehat{v}{\rm{d}}y,-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta }\right)}_{{\mathscr{G}}}={\left(\lambda \widehat{h},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}},since integrating (4.8)5{\left(4.8)}_{5}over G{\mathscr{G}}gives (1,θ^)G=0{\left(1,\widehat{\theta })}_{{\mathscr{G}}}=0due to λ≠0\lambda \ne 0. Noting the self-adjointness of ℬG{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}, we deduce that (u^,Φ)F=(u^,λ¯v^)F+(λℬGh^,θ^)G+2μ(D(u^),D(v^))F+(2ω(e3×u^),v^)F.{\left(\widehat{u},\Phi )}_{{\mathscr{F}}}={\left(\widehat{u},\overline{\lambda }\widehat{v})}_{{\mathscr{F}}}+{\left(\lambda {{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h},\widehat{\theta })}_{{\mathscr{G}}}+2\mu {\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}.Analogously, we obtain 0=(λu^−μΔu^+2ω(e3×u^)+∇q^,v^)F=(u^,λ¯v)F−((2μD(u^)−p^I)νG,v^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(−ℬGh^,v^⋅νG)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(−ℬGh^,λ¯θ^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(λh^,−ℬGθ^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F=(u^,λ¯v)F−(u^⋅νG,−ℬGθ^)G+2(D(u^),D(v^))F+(2ω(e3×u^),v^)F.\begin{array}{rcl}0& =& {\left(\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q},\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(\left(2\mu D\left(\widehat{u})-\widehat{p}I){\nu }_{{\mathscr{G}}},\widehat{v})}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h},\widehat{v}\cdot {\nu }_{{\mathscr{G}}})}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h},\overline{\lambda }\widehat{\theta })}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(\lambda \widehat{h},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}\\ & =& {\left(\widehat{u},\overline{\lambda }v)}_{{\mathscr{F}}}-{\left(\widehat{u}\cdot {\nu }_{{\mathscr{G}}},-{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{\theta })}_{{\mathscr{G}}}+2{\left(D\left(\widehat{u}),D\left(\widehat{v}))}_{{\mathscr{F}}}+{\left(2\omega \left({e}_{3}\times \widehat{u}),\widehat{v})}_{{\mathscr{F}}}.\end{array}Hence, we arrive at (u^,Φ)F=0{\left(\widehat{u},\Phi )}_{{\mathscr{F}}}=0for any Φ∈Cc∞(F)3\Phi \in {C}_{c}^{\infty }{\left({\mathscr{F}})}^{3}. Since Φ\Phi is arbitrary, we obtain u^=0\widehat{u}=0in F{\mathscr{F}}. Then, it follows from the equation (4.2)5{\left(4.2)}_{5}that h^=0\widehat{h}=0on G{\mathscr{G}}, as λ≠0\lambda \ne 0. This completes the proof.□Using the operator valued Fourier multiplier theorem due to Prüss [21], we find that Theorem 4.1 immediately follows from Lemma 4.2, see also the discussion in the proof of [39, Thm. 3.5].5Decay estimate for the linearized equationsIn this section, we shall address some exponential decay property of the linearized system (4.1). To this end, let {φm}m=14{\left\{{\varphi }_{m}\right\}}_{m=1}^{4}be an orthogonal basis of N(ℬG)∪C{\mathsf{N}}\left({{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}})\cup {\mathbb{C}}with respect to L2{L}^{2}inner-product (⋅,⋅)G{\left(\cdot ,\cdot )}_{{\mathscr{G}}}, where N(ℬG){\mathsf{N}}\left({{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}})stands for the null space of ℬG{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}. Here, φm{\varphi }_{m}, m=1,2,3,4m=1,2,3,4, are given by φ1=∣G∣−1/2{\varphi }_{1}=| {\mathscr{G}}{| }^{-1\text{/}2}, and φℓ+1=Cℓyℓ{\varphi }_{\ell +1}={C}_{\ell }{y}_{\ell }, ℓ=1,2,3\ell =1,2,3, respectively, normalized by (φj,φk)G=δjk{\left({\varphi }_{j},{\varphi }_{k})}_{{\mathscr{G}}}={\delta }_{jk}, where Cℓ{C}_{\ell }are constants. Assume that λ1{\lambda }_{1}is the same constant as in Lemma 4.2 in what follows. This section is dedicated to show the following theorem.Theorem 5.1Let 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Set J=(0,T)J=\left(0,T)with T>0T\gt 0. Then, there exists a constant ε0>0{\varepsilon }_{0}\gt 0such that the following assertion is valid: Let u0∈Bq,p2(δ−1/p)(F)3{u}_{0}\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}, h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), and (fu,gd,guτ,guv,fh)∈Fδ(J;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})that satisfy the compatibility conditions given in Theorem 4.1. The problem (4.1) admits a unique solution (u,q,TrG[q],h)∈Eδ(J;F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})possessing the estimate∣eε0t(u,q,TrG[q],h)∣Eδ(J;F)≤C∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)+∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p\begin{array}{l}| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left(J;{\mathscr{F}})}\\ \hspace{1.0em}\le C\left[| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right.\\ \hspace{1.0em}\hspace{1.0em}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}\left.+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right]\end{array}with some constant C independent of T.We first decompose (u,q,h)\left(u,q,h)of (4.1). To this end, we recall a unique solvability result of the weak Dirichlet problem: For any f∈Lq(F)3f\in {L}^{q}{\left({\mathscr{F}})}^{3}, 1<q<∞1\lt q\lt \infty , there exists a unique θ∈H˙01,q(F)\theta \in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}})satisfying (5.1)(∇θ,∇φ)F=(f,∇φ)Ffor any φ∈H˙01,q′(F){\left(\nabla \theta ,\nabla \varphi )}_{{\mathscr{F}}}={(f,\nabla \varphi )}_{{\mathscr{F}}}\hspace{1.0em}\hspace{0.1em}\text{for any\hspace{0.5em}}\hspace{0.1em}\varphi \in {\dot{H}}_{0}^{1,q^{\prime} }\hspace{0.33em}\left({\mathscr{F}})\text{}and possessing the estimate ∣∇θ∣Lq(F)≤C∣f∣Lq(F)| \nabla \theta {| }_{{L}^{q}\left({\mathscr{F}})}\le C| f{| }_{{L}^{q}\left({\mathscr{F}})}with a constant CCindependent of the choices of θ\theta , φ\varphi , and ff. Then, for any f∈Lq(F)3f\in {L}^{q}{\left({\mathscr{F}})}^{3}, we define the operators PF{{\mathbb{P}}}_{{\mathscr{F}}}and QF{{\mathbb{Q}}}_{{\mathscr{F}}}by ∇QFf≔∇θ\nabla {{\mathbb{Q}}}_{{\mathscr{F}}}f:= \nabla \theta and PFf≔(I−∇QF)f∈Jq(F){{\mathbb{P}}}_{{\mathscr{F}}}f:= \left(I-\nabla {{\mathbb{Q}}}_{{\mathscr{F}}})f\in {J}_{q}\left({\mathscr{F}})with θ∈H˙01,q(F)\theta \in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}})satisfying (5.1). Here, Jq(F){J}_{q}\left({\mathscr{F}})is the solenoidal space given by Jq(F)≔{f∈Lq(F)3:(f,∇φ)F=0for all φ∈H˙01,q′(F)}.{J}_{q}\left({\mathscr{F}}):= \{f\in {L}^{q}{\left({\mathscr{F}})}^{3}\hspace{0.33em}:\hspace{0.33em}{(f,\nabla \varphi )}_{{\mathscr{F}}}=0\hspace{1em}\hspace{0.1em}\text{for all\hspace{0.5em}}\hspace{0.1em}\varphi \in {\dot{H}}_{0}^{1,q^{\prime} }\hspace{0.33em}\left({\mathscr{F}})\text{}\}.Next, we consider the following systems: (5.2)∂tv∗+2λ2v∗−μΔv∗+2ω(e3×v∗)+∇π∗=fu,in F,divv∗=gd,in F,PG(2μD(v∗)νG)=guτ,on G,2μD(v∗)νG⋅νG−π∗+ℬGη∗=guv,on G,∂tη∗+2λ2η∗−(P0Gu∗)⋅νG=fh,on G,v∗(0)=u0,in F,η∗(0)=h0,on G.\left\{\begin{array}{ll}{\partial }_{t}{v}_{\ast }+2{\lambda }_{2}{v}_{\ast }-\mu \Delta {v}_{\ast }+2\omega \left({e}_{3}\times {v}_{\ast })+\nabla {\pi }_{\ast }={f}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{v}_{\ast }={g}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({v}_{\ast }){\nu }_{{\mathscr{G}}})={g}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({v}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{\pi }_{\ast }+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}{\eta }_{\ast }={g}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{\eta }_{\ast }+2{\lambda }_{2}{\eta }_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{u}_{\ast })\cdot {\nu }_{{\mathscr{G}}}={f}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {v}_{\ast }\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\eta }_{\ast }\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.(5.3)∂tu∗−μΔu∗+2ω(e3×u∗)+∇q∗=2λ2PFv˜∗,in F,divu∗=0,in F,PG(2μD(u∗)νG)=0,on G,2μD(u∗)νG⋅νG−q∗+ℬGh∗=0,on G,∂th∗−(P0Gu∗)⋅νG=2λ2η˜∗,on G,u∗(0)=0,in F,h∗(0)=0,on G.\left\{\begin{array}{ll}{\partial }_{t}{u}_{\ast }-\mu \Delta {u}_{\ast }+2\omega \left({e}_{3}\times {u}_{\ast })+\nabla {q}_{\ast }=2{\lambda }_{2}{{\mathbb{P}}}_{{\mathscr{F}}}{\widetilde{v}}_{\ast },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{u}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({u}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({u}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{q}_{\ast }+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}{h}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{h}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{u}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=2{\lambda }_{2}{\widetilde{\eta }}_{\ast },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {u}_{\ast }\left(0)=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {h}_{\ast }\left(0)=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Here, we have set v˜∗≔v∗−(v∗,1)F−∑α=1,2(v∗,eα×y)F−ω∫Gη˜∗yαy3dy(eα×y)−(v∗,e3×y)F(e3×y){\widetilde{v}}_{\ast }:= {v}_{\ast }-{\left({v}_{\ast },1)}_{{\mathscr{F}}}-\sum _{\alpha =1,2}\left({\left({v}_{\ast },{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}{\widetilde{\eta }}_{\ast }{y}_{\alpha }{y}_{3}{\rm{d}}y\right)\left({e}_{\alpha }\times y)-{\left({v}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}\left({e}_{3}\times y)and η˜∗≔η∗−∑m=14(η∗,φm)Gφm{\widetilde{\eta }}_{\ast }:= {\eta }_{\ast }-{\sum }_{m=1}^{4}{\left({\eta }_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}{\varphi }_{m}, respectively. Now, setting (5.4)u≔v∗+u∗+2λ2∫0t(v∗(s),1)F+∑α=1,2(v∗(s),eα×y)F−ω∫Gη˜∗(s)yαy3dy(eα×y)+(v∗(s),e3×y)F(e3×y)ds,u:= {v}_{\ast }+{u}_{\ast }+2{\lambda }_{2}\underset{0}{\overset{t}{\int }}\left\{{\left({v}_{\ast }\left(s),1)}_{{\mathscr{F}}}+\sum _{\alpha =1,2}\left({\left({v}_{\ast }\left(s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}{\widetilde{\eta }}_{\ast }\left(s){y}_{\alpha }{y}_{3}{\rm{d}}y\right)\left({e}_{\alpha }\times y)+{\left({v}_{\ast }\left(s),{e}_{3}\times y)}_{{\mathscr{F}}}\left({e}_{3}\times y)\right\}{\rm{d}}s,and (5.5)q≔π∗+q∗−2λ2∇QFv˜∗,h≔η∗+h∗+2λ2∑m=14∫0t(η∗(s),φm)Gφmds,q:= {\pi }_{\ast }+{q}_{\ast }-2{\lambda }_{2}\nabla {{\mathbb{Q}}}_{{\mathscr{F}}}{\widetilde{v}}_{\ast },\hspace{1.0em}h:= {\eta }_{\ast }+{h}_{\ast }+2{\lambda }_{2}\mathop{\sum }\limits_{m=1}^{4}\underset{0}{\overset{t}{\int }}{\left({\eta }_{\ast }\left(s),{\varphi }_{m})}_{{\mathscr{G}}}{\varphi }_{m}{\rm{d}}s,we see that (u,q,h)\left(u,q,h)is a solution to (4.1). In the following, we construct the solutions of (5.2) and (5.3), respectively.Step 1: Study of (5.2). For the shifted system (5.2), we have the next theorem.Theorem 5.2Assume that 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Set J=(0,T)J=\left(0,T)with T>0T\gt 0. Let u0∈Bq,p2(δ−1/p)(F)3{u}_{0}\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}, h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), and (fu,gd,guτ,guv,fh)∈Fδ(J;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})that satisfy the compatibility conditions given in Theorem 4.1. Then, for any λ2>λ1/2{\lambda }_{2}\gt {\lambda }_{1}\hspace{0.1em}\text{/}\hspace{0.1em}2, system (5.2) has a unique solution (v∗,π∗,TrG[π∗],η∗)∈Eδ(J;F)\left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{G}}}\left[{\pi }_{\ast }],{\eta }_{\ast })\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})possessing the estimate∣(v∗,π∗,TrF[π∗],η∗)∣Eδ(J;F)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣(fu,gd,guτ,guv,fh)∣Fδ(J;F))| \left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{F}}}\left[{\pi }_{\ast }],{\eta }_{\ast }){| }_{{{\mathbb{E}}}_{\delta }\left(J;{\mathscr{F}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| ({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})for some constant C independent of TT, ω\omega , λ1{\lambda }_{1}, and λ2{\lambda }_{2}.ProofFor λ∈Σε,λ1\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{1}}, let Uω(λ){{\mathcal{U}}}_{\omega }\left(\lambda ), Qω(λ){{\mathcal{Q}}}_{\omega }\left(\lambda ), and ℋω(λ){{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda )be operators given in Lemma 4.2. Let λ2{\lambda }_{2}and ε1{\varepsilon }_{1}be numbers for which 2λ2−λ1>ε1>02{\lambda }_{2}-{\lambda }_{1}\gt {\varepsilon }_{1}\gt 0. Then, for every λ∈−ε1+Σε,0\lambda \in -{\varepsilon }_{1}+{\Sigma }_{\varepsilon ,0}, it holds λ+2λ2∈λ1+Σε,0\lambda +2{\lambda }_{2}\in {\lambda }_{1}+{\Sigma }_{\varepsilon ,0}. Hence, by Lemma 4.2, we obtain ℛXq→H2−j,q(F)3({(τ∂τ)ℓ(λj/2Uω(λ+2λ2)):−ε1+λ∈Σε,0})≤c,ℛXq→Lq(F)3({(τ∂τ)ℓ∇Qω(λ+2λ2):−ε1+λ∈Σε,0})≤c,ℛXq→Bq,q3−1/q−k(G)({(τ∂τ)ℓ(λkℋω(λ+2λ2)):−ε1+λ∈Σε,0})≤c\begin{array}{rcl}{{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {H}^{2-j,q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{j\text{/}2}{{\mathcal{U}}}_{\omega }\left(\lambda +2{\lambda }_{2}))\hspace{0.33em}:\hspace{0.33em}-{\varepsilon }_{1}+\lambda \in {\Sigma }_{\varepsilon ,0}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {L}^{q}{\left({\mathscr{F}})}^{3}}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\nabla {{\mathcal{Q}}}_{\omega }\left(\lambda +2{\lambda }_{2})\hspace{0.33em}:\hspace{0.33em}-{\varepsilon }_{1}+\lambda \in {\Sigma }_{\varepsilon ,0}\})& \le & c,\\ {{\mathcal{ {\mathcal R} }}}_{{{\mathcal{X}}}_{q}\to {B}_{q,q}^{3-1\text{/}q-k}\left({\mathscr{G}})}(\{{\left(\tau {\partial }_{\tau })}^{\ell }\left({\lambda }^{k}{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda +2{\lambda }_{2}))\hspace{0.33em}:\hspace{0.33em}-{\varepsilon }_{1}+\lambda \in {\Sigma }_{\varepsilon ,0}\})& \le & c\end{array}for ℓ=0,1\ell =0,1, j=0,1,2j=0,1,2, and k=0,1k=0,1. Employing the argument in the proof of [39, Thm. 3.5] readily implies the required assertion. This completes the proof.□For every ε0>0{\varepsilon }_{0}\gt 0, we see that eε0t(v∗,π∗,η∗){e}^{{\varepsilon }_{0}t}\left({v}_{\ast },{\pi }_{\ast },{\eta }_{\ast })satisfies ∂t(eε0tv∗)+(2λ2−ε0)(eε0v∗)−μΔ(eε0tv∗)+2ω(e3×(eε0tv∗))+∇(eε0tπ∗)=eε0tfu,in F,div(eε0tv∗)=eε0tgd,in F,PG(2μD(eε0tv∗)νG)=eε0tguτ,on G,2μD(eε0tv∗)νG⋅νG−(eε0tπ∗)+ℬG(eε0tη∗)=eε0tguv,on G,∂t(eε0tη∗)+(2λ2−ε0)(eε0tη∗)−(P0G(eε0tv∗))⋅νG=eε0tfh,on G,v∗(0)=u0,in F,η∗(0)=h0,on G.\left\{\begin{array}{ll}{\partial }_{t}\left({e}^{{\varepsilon }_{0}t}{v}_{\ast })+\left(2{\lambda }_{2}-{\varepsilon }_{0})\left({e}^{{\varepsilon }_{0}}{v}_{\ast })-\mu \Delta \left({e}^{{\varepsilon }_{0}t}{v}_{\ast })+2\omega \left({e}_{3}\times \left({e}^{{\varepsilon }_{0}t}{v}_{\ast }))+\nabla \left({e}^{{\varepsilon }_{0}t}{\pi }_{\ast })={e}^{{\varepsilon }_{0}t}{f}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\left({e}^{{\varepsilon }_{0}t}{v}_{\ast })={e}^{{\varepsilon }_{0}t}{g}_{d},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({e}^{{\varepsilon }_{0}t}{v}_{\ast }){\nu }_{{\mathscr{G}}})={e}^{{\varepsilon }_{0}t}{g}_{u\tau },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({e}^{{\varepsilon }_{0}t}{v}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\left({e}^{{\varepsilon }_{0}t}{\pi }_{\ast })+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\left({e}^{{\varepsilon }_{0}t}{\eta }_{\ast })={e}^{{\varepsilon }_{0}t}{g}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}\left({e}^{{\varepsilon }_{0}t}{\eta }_{\ast })+\left(2{\lambda }_{2}-{\varepsilon }_{0})\left({e}^{{\varepsilon }_{0}t}{\eta }_{\ast })-\left({P}_{0}^{{\mathscr{G}}}\left({e}^{{\varepsilon }_{0}t}{v}_{\ast }))\cdot {\nu }_{{\mathscr{G}}}={e}^{{\varepsilon }_{0}t}{f}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {v}_{\ast }\left(0)={u}_{0},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\eta }_{\ast }\left(0)={h}_{0},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.\end{array}\right.Given ε0>0{\varepsilon }_{0}\gt 0, we choose λ2>0{\lambda }_{2}\gt 0suitably large such that 2λ2−λ1>ε0>02{\lambda }_{2}-{\lambda }_{1}\gt {\varepsilon }_{0}\gt 0. Then, from Theorem 5.2, we obtain the following corollary, which gives the decay property of solutions to (5.2).Corollary 5.3Let 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Set J=(0,T)J=\left(0,T)with T>0T\gt 0. Let u0∈Bq,p2(δ−1/p)(F)3{u}_{0}\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}, h0∈Bq,p2+δ−1/p−1/q(G){h}_{0}\in {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}), and (fu,gd,guτ,guv,fh)∈Fδ(J;F)({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h})\in {{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})that satisfy the compatibility conditions given in Theorem 4.1. There exists a constant λ2>0{\lambda }_{2}\gt 0such that the system (5.2) admits a unique solution (v∗,π∗,TrG[π∗],η∗)∈Eδ(J;F)\left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{G}}}\left[{\pi }_{\ast }],{\eta }_{\ast })\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})possessing the estimate∣eε0t(v∗,π∗,TrG[π∗],η∗)∣Eδ(J;F)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F))| {e}^{{\varepsilon }_{0}t}\left({v}_{\ast },{\pi }_{\ast },{{\rm{Tr}}}_{{\mathscr{G}}}\left[{\pi }_{\ast }],{\eta }_{\ast }){| }_{{{\mathbb{E}}}_{\delta }\left(J;{\mathscr{F}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})with a constant C independent of TT, ω\omega , λ1{\lambda }_{1}, and λ2{\lambda }_{2}, where ε0{\varepsilon }_{0}is a number such that 0<ε0<2λ2−λ10\lt {\varepsilon }_{0}\lt 2{\lambda }_{2}-{\lambda }_{1}.Step 2: Study of (5.3). To study (5.3), we rely on the following decomposition: (5.6)u∗(y,t)=U∗(y,t)+d3[h∗](e3×y),q∗(y,t)=Q∗(y,t)+ωd3[h∗]∣y′∣2+1∣G∣∫GCGh∗dG,{u}_{\ast }(y,t)={U}_{\ast }(y,t)+{d}_{3}\left[{h}_{\ast }]\left({e}_{3}\times y),\hspace{1.0em}{q}_{\ast }(y,t)={Q}_{\ast }(y,t)+\omega {d}_{3}\left[{h}_{\ast }]| y^{\prime} {| }^{2}+\frac{1}{| {\mathscr{G}}| }\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }{\rm{d}}{\mathscr{G}},where we have set dℓ[h∗]≔−ωSℓ∫Gh∗(y,t)(e3×y)⋅(eℓ×y)dG,Sℓ≔∣eℓ×y∣L2(F)2=∫F(∣y∣2−yℓ2)dy,CGh∗≔ℬGh∗−ωd3[h∗]∣y′∣2\begin{array}{rcl}{d}_{\ell }\left[{h}_{\ast }]& := & -\frac{\omega }{{{\mathcal{S}}}_{\ell }}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }(y,t)\left({e}_{3}\times y)\cdot \left({e}_{\ell }\times y){\rm{d}}{\mathscr{G}},\\ {{\mathcal{S}}}_{\ell }& := & | {e}_{\ell }\times y{| }_{{L}^{2}\left({\mathscr{F}})}^{2}=\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left(| y{| }^{2}-{y}_{\ell }^{2}){\rm{d}}y,\\ {{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }& := & {{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}{h}_{\ast }-\omega {d}_{3}\left[{h}_{\ast }]| y^{\prime} {| }^{2}\end{array}with ℓ=1,2,3\ell =1,2,3. We further set C^Gh∗≔CGh∗−1∣G∣∫GCGh∗dG.{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }:= {{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }-\frac{1}{| {\mathscr{G}}| }\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{C}}}_{{\mathscr{G}}}{h}_{\ast }{\rm{d}}{\mathscr{G}}.Noting that 2e3×(e3×y)=−(2y1,2y2,0)=−∇∣y′∣2,∂td3[h∗]=−ωS3∫G∂th∗∣y′∣2dG=−ωS3∫G((P0GU∗)⋅νG+2λ2η˜∗)∣y′∣2dG,\begin{array}{rcl}2{e}_{3}\times \left({e}_{3}\times y)& =& -\left(2{y}_{1},2{y}_{2},0)=-\nabla | y^{\prime} {| }^{2},\\ {\partial }_{t}{d}_{3}\left[{h}_{\ast }]& =& -\frac{\omega }{{{\mathcal{S}}}_{3}}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\partial }_{t}{h}_{\ast }| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}=-\frac{\omega }{{{\mathcal{S}}}_{3}}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}(\left({P}_{0}^{{\mathscr{G}}}{U}_{\ast })\cdot {\nu }_{{\mathscr{G}}}+2{\lambda }_{2}{\widetilde{\eta }}_{\ast })| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}},\end{array}the system (5.3) can be rewritten as follows: (5.7)∂tU∗−Lω,yU∗+∇Q∗=2λ2f˜w,in F,divU∗=0,in F,PG(2μD(U∗)νG)=0,on G,2μD(U∗)νG⋅νG−Q∗+C^Gh∗=0,on G,∂th∗−(P0GU∗)⋅νG=2λ2η˜∗,on G,U∗(0)=0,in F,h∗(0)=0,on G,\left\{\begin{array}{ll}{\partial }_{t}{U}_{\ast }-{L}_{\omega ,y}{U}_{\ast }+\nabla {Q}_{\ast }=2{\lambda }_{2}{\widetilde{f}}_{w},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{U}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({U}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{Q}_{\ast }+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{h}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{U}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=2{\lambda }_{2}{\widetilde{\eta }}_{\ast },& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {U}_{\ast }\left(0)=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {h}_{\ast }\left(0)=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where we have set Lω,yw≔2ωS3∫F(P0Gw)⋅y′dy(e3×y)+μΔw−2ω(e3×w),f˜w≔PFv˜∗+ωS3∫Gη˜∗∣y′∣2dG(e3×y).\begin{array}{rcl}{L}_{\omega ,y}w& := & \frac{2\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({P}_{0}^{{\mathscr{G}}}w)\cdot y^{\prime} {\rm{d}}y\right)\left({e}_{3}\times y)+\mu \Delta w-2\omega \left({e}_{3}\times w),\\ {\widetilde{f}}_{w}& := & {{\mathbb{P}}}_{{\mathscr{F}}}{\widetilde{v}}_{\ast }+\frac{\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widetilde{\eta }}_{\ast }| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)\left({e}_{3}\times y).\end{array}Notice that it holds (e3×y)⋅νG=0\left({e}_{3}\times y)\cdot {\nu }_{{\mathscr{G}}}=0on G{\mathscr{G}}due to the axisymmetry of G{\mathscr{G}}.As a base space for our analysis, we use X0=Jq(F)×Bq,q2−1/q(G),{X}_{0}={J}_{q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}),and we set X¯1≔H2,q(F)3×Bq,q3−1/q(G).{\overline{X}}_{1}:= {H}^{2,q}{\left({\mathscr{F}})}^{3}\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}).Define a closed linear operator in X0{X}_{0}by means of Aq(U∗,h∗)≔(−Lω,yU∗+∇K(U∗,h∗),−(P0GU∗)⋅νG){{\mathcal{A}}}_{q}\left({U}_{\ast },{h}_{\ast }):= \left(-{L}_{\omega ,y}{U}_{\ast }+\nabla K\left({U}_{\ast },{h}_{\ast }),-\left({P}_{0}^{{\mathscr{G}}}{U}_{\ast })\cdot {\nu }_{{\mathscr{G}}})with domain X1≔D(Aq)⊂X¯1{X}_{1}:= {\mathsf{D}}\left({{\mathcal{A}}}_{q})\subset {\overline{X}}_{1}defined by D(Aq)≔{(U∗,h∗)∈X0∩X¯1:PG(2μD(U∗)νG)=0on G}.{\mathsf{D}}\left({{\mathcal{A}}}_{q}):= \left\{\left({U}_{\ast },{h}_{\ast })\in {X}_{0}\cap {\overline{X}}_{1}\hspace{0.33em}:\hspace{0.33em}{{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})=0\hspace{0.33em}\hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\right\}.Here, K(U∗,h∗)∈H˙01,q(F)K\left({U}_{\ast },{h}_{\ast })\in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}})is a functional that is a unique solution to the weak Dirichlet problem (5.8)(∇K(U∗,h∗),∇φ)F=(Lω,yU∗,∇φ)Ffor any φ∈H˙01,q′(F),K(U∗,h∗)=2μ(D(U∗)νG)⋅νG+C^Gh∗on G\left\{\begin{array}{ll}{\left(\nabla K\left({U}_{\ast },{h}_{\ast }),\nabla \varphi )}_{{\mathscr{F}}}={\left({L}_{\omega ,y}{U}_{\ast },\nabla \varphi )}_{{\mathscr{F}}}& \hspace{0.1em}\text{for any\hspace{0.5em}}\hspace{0.1em}\varphi \in {\dot{H}}_{0}^{1,q^{\prime} }\left({\mathscr{F}})\text{},\\ K\left({U}_{\ast },{h}_{\ast })=2\mu \left(D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})\cdot {\nu }_{{\mathscr{G}}}+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for (U∗,h∗)∈X0∩X¯1\left({U}_{\ast },{h}_{\ast })\in {X}_{0}\cap {\overline{X}}_{1}. Since (U∗,h∗)\left({U}_{\ast },{h}_{\ast })belongs to X0∩X¯1{X}_{0}\cap {\overline{X}}_{1}, we have ∣Lω,yU∗∣Lq(F)≤Cq,F(μ∣U∗∣H2,q(F)+ω∣U∗∣Lq(F)),∣2μ(D(U∗)νG)⋅νG+C^Gh∗∣Bq,q1−1/q(G)≤Cq,G(μ∣U∗∣H2,q(F)+σ∣h∗∣Bq,q3−1/q(G)+ω2∣h∗∣Bq,q2−1/q(G)),\begin{array}{rcl}| {L}_{\omega ,y}{U}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}& \le & {C}_{q,{\mathscr{F}}}(\mu | {U}_{\ast }{| }_{{H}^{2,q}\left({\mathscr{F}})}+\omega | {U}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}),\\ | 2\mu \left(D\left({U}_{\ast }){\nu }_{{\mathscr{G}}})\cdot {\nu }_{{\mathscr{G}}}+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{h}_{\ast }{| }_{{B}_{q,q}^{1-1\text{/}q}\left({\mathscr{G}})}& \le & {C}_{q,{\mathscr{G}}}(\mu | {U}_{\ast }{| }_{{H}^{2,q}\left({\mathscr{F}})}+\sigma | {h}_{\ast }{| }_{{B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}})}+{\omega }^{2}| {h}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}),\end{array}which yields the estimate for ∇K(U∗,h∗)\nabla K\left({U}_{\ast },{h}_{\ast }): ∣∇K(U∗,h∗)∣Lq(F)≤Cq,G(μ∣U∗∣H2,q(F)+ω∣U∗∣Lq(F)+σ∣h∗∣Bq,q3−1/q(G)+ω2∣h∗∣Bq,q2−1/q(G)).| \nabla K\left({U}_{\ast },{h}_{\ast }){| }_{{L}^{q}\left({\mathscr{F}})}\le {C}_{q,{\mathscr{G}}}(\mu | {U}_{\ast }{| }_{{H}^{2,q}\left({\mathscr{F}})}+\omega | {U}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+\sigma | {h}_{\ast }{| }_{{B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}})}+{\omega }^{2}| {h}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}).As F{\mathscr{F}}is a bounded smooth domain, the weak Dirichlet problem (5.8) admits a unique solution K(U∗,h∗)∈H˙01,q(F)K\left({U}_{\ast },{h}_{\ast })\in {\dot{H}}_{0}^{1,q}\left({\mathscr{F}}), i.e., the functional K(U∗,h∗)K\left({U}_{\ast },{h}_{\ast })is well defined. Notice that the solution K(U∗,h∗)K\left({U}_{\ast },{h}_{\ast })to (5.8) depends on ω\omega . Applying the similar argument in [25, Sec. 9.2.1], we see that the system (5.7) is equivalent to the abstract evolution equation z˙+Aqz=(2λ2f˜w,2λ2η˜∗),t>0,z(0)=0\dot{z}+{{\mathcal{A}}}_{q}z\left=(2{\lambda }_{2}{\widetilde{f}}_{w}\left,2{\lambda }_{2}{\widetilde{\eta }}_{\ast }),\hspace{1.0em}t\gt 0,\hspace{1.0em}z\left(0)=0with z=(U∗,h∗)z=\left({U}_{\ast },{h}_{\ast }). Employing the standard Neumann series argument, we can prove that the operator −Aq-{{\mathcal{A}}}_{q}generates an analytic C0{C}_{0}-semigroup in X0{X}_{0}.Lemma 5.4Let 1<q<∞1\lt q\lt \infty . Then −Aq-{{\mathcal{A}}}_{q}with domain D(Aq){\mathsf{D}}\left({{\mathcal{A}}}_{q})generates an analytic C0{C}_{0}-semigroup in X0{X}_{0}.ProofLet us consider the following resolvent problem: (5.9)λU^∗−Lω,yU^∗+∇Q^∗=f^u,in F,divU^∗=0,in F,PG(2μD(U^∗)νG)=0,on G,2μD(U^∗)νG⋅νG−Q^∗+C^Gh^∗=g^uv,on G,λh^∗−(P0GU^∗)⋅νG=f^h,on G\left\{\begin{array}{ll}\lambda {\widehat{U}}_{\ast }-{L}_{\omega ,y}{\widehat{U}}_{\ast }+\nabla {\widehat{Q}}_{\ast }={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{\widehat{U}}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{\widehat{Q}}_{\ast }+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast }={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for any λ∈Σε,λ4\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{4}}and (f^u,0,0,g^uv,f^h)∈Xq,σ({\widehat{f}}_{u},0,0,{\widehat{g}}_{uv},{\widehat{f}}_{h})\in {X}_{q,\sigma }with some positive constant λ4{\lambda }_{4}, where we have set Xq,σ≔Xq∩(Jq(F)×DIq(F)×H1,q(F)2×H1,q(F)×Bq,q2−1/q(G)).{X}_{q,\sigma }:= {X}_{q}\cap ({J}_{q}\left({\mathscr{F}})\times {{\rm{DI}}}_{q}\left({\mathscr{F}})\times {H}^{1,q}{\left({\mathscr{F}})}^{2}\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})).To find solutions of the aforementioned problem, let us consider (5.10)λu^−μΔu^+2ω(e3×u^)+∇q^=f^u,in F,divu^=0,in F,PG(2μD(u^)νG)=0,on G,2μD(u^)νG⋅νG−q^+ℬGh^=g^uv,on G,λh^−(P0Gu^)⋅νG=f^h,on G\left\{\begin{array}{ll}\lambda \widehat{u}-\mu \Delta \widehat{u}+2\omega \left({e}_{3}\times \widehat{u})+\nabla \widehat{q}={\widehat{f}}_{u},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widehat{q}+{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}\widehat{h}={\widehat{g}}_{uv},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for (f^u,0,0,g^uv,f^h)∈Xq,σ({\widehat{f}}_{u},0,0,{\widehat{g}}_{uv},{\widehat{f}}_{h})\in {X}_{q,\sigma }and λ∈Σε,λ2\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{2}}. According to Lemma 4.2, there exists a unique solution (u^,q^,h^)∈(H2,q(F)3∩Jq(F))×H1,q(F)×Bq,q3−1/q(G)\left(\widehat{u},\widehat{q},\widehat{h})\in \left({H}^{2,q}{\left({\mathscr{F}})}^{3}\cap {J}_{q}\left({\mathscr{F}}))\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})of (5.10). Let Uω{{\mathcal{U}}}_{\omega }and ℋω{{\mathcal{ {\mathcal H} }}}_{\omega }be the operators given in Lemma 4.2. If we set q˜≔q^−∣G∣−1∫GCGh^dG\widetilde{q}:= \widehat{q}-| {\mathscr{G}}{| }^{-1}{\int }_{{\mathscr{G}}}{{\mathcal{C}}}_{{\mathscr{G}}}\widehat{h}{\rm{d}}{\mathscr{G}}, for λ∈Σε,λ2\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{2}}, we see that (u^,q˜,h^)\left(\widehat{u},\widetilde{q},\widehat{h})satisfies λu^−Lω,yu^+∇q˜=f^u+2ωS3∫F(P0GUω(λ)F0)⋅y′dy(e3×y),in F,divu^=0,in F,PG(2μD(u^)νG)=0,on G,2μD(u^)νG⋅νG−q˜+C^Gh^=f^uv+ωd3[ℋω(λ)F0]∣y′∣2,on G,λh^−(P0Gu^)⋅νG=f^h,on G,\left\{\begin{array}{ll}\lambda \widehat{u}-{L}_{\omega ,y}\widehat{u}+\nabla \widetilde{q}={\widehat{f}}_{u}+\frac{2\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({P}_{0}^{{\mathscr{G}}}{{\mathcal{U}}}_{\omega }\left(\lambda ){F}_{0})\cdot y^{\prime} {\rm{d}}y\right)\left({e}_{3}\times y),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}\widehat{u}=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widehat{u}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-\widetilde{q}+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widehat{h}={\widehat{f}}_{uv}+\omega {d}_{3}\left[{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ){F}_{0}]| y^{\prime} {| }^{2},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widehat{h}-\left({P}_{0}^{{\mathscr{G}}}\widehat{u})\cdot {\nu }_{{\mathscr{G}}}={\widehat{f}}_{h},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.where we have set F0=(f^u,0,⋯,0,λ1/2g^uv,g^uv,f^h)∈Xq{F}_{0}=({\widehat{f}}_{u},0,\cdots \hspace{0.33em},0,{\lambda }^{1\text{/}2}{\widehat{g}}_{uv},{\widehat{g}}_{uv},{\widehat{f}}_{h})\in {{\mathcal{X}}}_{q}. Since it holds 2ωS3∫F(P0GUω(λ)F0)⋅y′dy(e3×y)Lq(F)≤C1ωλ3−1∣F0∣Xq,∣ωd3[ℋω(λ)F0]∣y′∣2∣Bq,q2−1/q(G)≤C1ω2λ3−1∣F0∣Xq\begin{array}{rcl}{\left|\frac{2\omega }{{{\mathcal{S}}}_{3}}\left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({P}_{0}^{{\mathscr{G}}}{{\mathcal{U}}}_{\omega }\left(\lambda ){F}_{0})\cdot y^{\prime} {\rm{d}}y\right)\left({e}_{3}\times y)\right|}_{{L}^{q}\left({\mathscr{F}})}& \le & {C}_{1}\omega {\lambda }_{3}^{-1}| {F}_{0}{| }_{{{\mathcal{X}}}_{q}},\\ | \omega {d}_{3}\left[{{\mathcal{ {\mathcal H} }}}_{\omega }\left(\lambda ){F}_{0}]| y^{\prime} {| }^{2}{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}& \le & {C}_{1}{\omega }^{2}{\lambda }_{3}^{-1}| {F}_{0}{| }_{{{\mathcal{X}}}_{q}}\end{array}for any λ3≥λ2{\lambda }_{3}\ge {\lambda }_{2}, we shall take λ3{\lambda }_{3}sufficiently large such that λ3≥4C1max(ω2,1){\lambda }_{3}\ge 4{C}_{1}\max \left({\omega }^{2},1)so that it follows from the Neumann series argument that (5.9) admits a unique solution (U^∗,Q^∗,h^∗)∈(H2,q(F)3∩Jq(F))×H1,q(F)×Bq,q3−1/q(G)\left({\widehat{U}}_{\ast },{\widehat{Q}}_{\ast },{\widehat{h}}_{\ast })\in \left({H}^{2,q}{\left({\mathscr{F}})}^{3}\cap {J}_{q}\left({\mathscr{F}}))\times {H}^{1,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})with λ4≥λ3{\lambda }_{4}\ge {\lambda }_{3}, where the resolvent estimate ∣(∣λ∣U^∗,∣λ∣1/2∇U^∗,∇2U^∗)∣Lq(F)+∣∇Q^∗∣Lq(F)+∣(∣λ∣h^∗,∇h^∗)∣Bq,q2−1/q(G)≤C(∣f^u∣Lq(F)+∣λ∣1/2∣f^uv∣Lq(F)+∣f^uv∣H1,q(F)+∣f^h∣Bq,q2−1/q(G))| \left(| \lambda | {\widehat{U}}_{\ast },| \lambda {| }^{1\text{/}2}\nabla {\widehat{U}}_{\ast },{\nabla }^{2}{\widehat{U}}_{\ast }){| }_{{L}^{q}\left({\mathscr{F}})}+| \nabla {\widehat{Q}}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+| \left(| \lambda | {\widehat{h}}_{\ast },\nabla {\widehat{h}}_{\ast }){| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}\le C(| {\widehat{f}}_{u}{| }_{{L}^{q}\left({\mathscr{F}})}+| \lambda {| }^{1\text{/}2}| {\widehat{f}}_{uv}{| }_{{L}^{q}\left({\mathscr{F}})}+| {\widehat{f}}_{uv}{| }_{{H}^{1,q}\left({\mathscr{F}})}+| {\widehat{f}}_{h}{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})})is valid with a constant CCdepending only on qq, G{\mathscr{G}}, μ\mu , and σ\sigma . If we set f^uv≡0{\widehat{f}}_{uv}\equiv 0, the resolvent problem λz^+Aqz^=f^with z^=(U^∗,h^∗)  and f^=(f^u,f^h)\lambda \widehat{z}+{{\mathcal{A}}}_{q}\widehat{z}=\widehat{f}\hspace{1.0em}\hspace{0.1em}\text{with\hspace{0.5em}}\hspace{0.1em}\widehat{z}=\left({\widehat{U}}_{\ast },{\widehat{h}}_{\ast })\hspace{0.1em}\text{\hspace{0.5em} and\hspace{0.5em}}\hspace{0.1em}\widehat{f}=({\widehat{f}}_{u},{\widehat{f}}_{h})\text{}is equivalent to (5.9) with Q^∗=K(U^∗,h^∗){\widehat{Q}}_{\ast }=K\left({\widehat{U}}_{\ast },{\widehat{h}}_{\ast })and admits a unique solution z^∈X1\widehat{z}\in {X}_{1}satisfying the resolvent estimate ∣λ∣∣z^∣X0+∣z^∣X¯1≤C∣f^∣X0,λ∈Σε,λ4.| \lambda | | \widehat{z}{| }_{{X}_{0}}+| \widehat{z}{| }_{{\overline{X}}_{1}}\le C| \widehat{f}{| }_{{X}_{0}},\hspace{1.0em}\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{4}}.Finally, the closedness of Aq{{\mathcal{A}}}_{q}follows from the fact that the resolvent set is not empty. This completes the proof.□To show the exponential decay property of the system (5.7), we introduce the following functional spaces: Let B˜q,q2−1/q(G){\widetilde{B}}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})be the set of all g∈Bq,q2−1/q(G)g\in {B}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfying (g,φm)G=0{\left(g,{\varphi }_{m})}_{{\mathscr{G}}}=0for m=1,2,3,4m=1,2,3,4, i.e., ∫GgdG=∫GgyℓdG=0,(ℓ=1,2,3).\mathop{\int }\limits_{{\mathscr{G}}}g{\rm{d}}{\mathscr{G}}=\mathop{\int }\limits_{{\mathscr{G}}}g{y}_{\ell }{\rm{d}}{\mathscr{G}}=0,\hspace{1.0em}\left(\ell =1,2,3).The subspace J˜q(F){\widetilde{J}}_{q}\left({\mathscr{F}})of Jq(F){J}_{q}\left({\mathscr{F}})stands for the set of all f∈Jq(F)f\in {J}_{q}\left({\mathscr{F}})satisfying the orthogonal conditions ∫Ffdy=∫Ff⋅(e3×y)dy=0,∫Ff⋅(eα×y)dy=ω∫Gg˜yαy3dG,(α=1,2),\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f{\rm{d}}y& =& \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{3}\times y){\rm{d}}y=0,\\ \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}f\cdot \left({e}_{\alpha }\times y){\rm{d}}y& =& \omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{g}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},\hspace{1.0em}\left(\alpha =1,2),\end{array}where g˜∈B˜q,q2−1/q(G)\widetilde{g}\in {\widetilde{B}}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}). We then define X˜0≔J˜q(F)∩B˜q,q2−1/q(G).{\widetilde{X}}_{0}:= {\widetilde{J}}_{q}\left({\mathscr{F}})\cap {\widetilde{B}}_{q,q}^{2-1\hspace{0.1em}\text{/}\hspace{0.1em}q}\left({\mathscr{G}}).We set the restriction operator A˜q≔Aq∣X˜0{\widetilde{{\mathcal{A}}}}_{q}:= {{\mathcal{A}}}_{q}{| }_{{\widetilde{X}}_{0}}with its domain given by D(A˜q)≔D(Aq)∩X˜0{\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q}):= {\mathsf{D}}\left({{\mathcal{A}}}_{q})\cap {\widetilde{X}}_{0}. Then the operator −A˜q-{\widetilde{{\mathcal{A}}}}_{q}generates an analytic C0{C}_{0}-semigroup on X˜0{\widetilde{X}}_{0}.Lemma 5.5Let 1<q<∞1\lt q\lt \infty . The induced operator −A˜q≔−Aq∣X˜0-{\widetilde{{\mathcal{A}}}}_{q}:= -{{\mathcal{A}}}_{q}{| }_{{\widetilde{X}}_{0}}with domain D(A˜q)≔D(Aq)∩X˜0{\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q}):= {\mathsf{D}}\left({{\mathcal{A}}}_{q})\cap {\widetilde{X}}_{0}is the generator of an analytic C0{C}_{0}-semigroup in X˜0{\widetilde{X}}_{0}.To prove this lemma, we need the following proposition given in [33, Prop. 2.3].Proposition 5.6Let rrbe a function defined on G{\mathscr{G}}and Gr{{\mathscr{G}}}_{r}be a normal perturbation of G{\mathscr{G}}given byGr≔{s=y+rνG(y):y∈G},{{\mathscr{G}}}_{r}:= \left\{s=y+r{\nu }_{{\mathscr{G}}}(y)\hspace{0.33em}:\hspace{0.33em}y\in {\mathscr{G}}\right\},where ∣r∣L∞(G)| r{| }_{{L}^{\infty }\left({\mathscr{G}})}and ∣∇Gr∣L∞(G)| {\nabla }_{{\mathscr{G}}}r{| }_{{L}^{\infty }\left({\mathscr{G}})}are assumed to be suitably small such that Gr{{\mathscr{G}}}_{r}is contained in the tubular neighborhood of G{\mathscr{G}}. For arbitrary function ζ(y)=a+b×y\zeta (y)=a+b\times y(with constants aaand bb) defined on G{\mathscr{G}}, the equality∫GC^GrνG⋅ζdG=−ω2∫Grζ⋅y′dG\mathop{\int }\limits_{{\mathscr{G}}}{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}r{\nu }_{{\mathscr{G}}}\cdot \zeta {\rm{d}}{\mathscr{G}}=-{\omega }^{2}\mathop{\int }\limits_{{\mathscr{G}}}r\zeta \cdot y^{\prime} {\rm{d}}{\mathscr{G}}holds.ProofThere are only a few words for the proof in [33], but we record the proof here for the reader’s convenience. We denote by Fr{{\mathscr{F}}}_{r}the closed domain surrounded by Gr{{\mathscr{G}}}_{r}. We consider the integral (5.11)I[r]≔∫GrσHGr+ω22∣s′∣2+p0νGr⋅ζdGr,I\left[r]:= \mathop{\int }\limits_{{{\mathscr{G}}}_{r}}\left(\sigma {{\mathscr{H}}}_{{{\mathscr{G}}}_{r}}+\frac{{\omega }^{2}}{2}| s^{\prime} {| }^{2}+{p}_{0}\right){\nu }_{{{\mathscr{G}}}_{r}}\cdot \zeta {\rm{d}}{{\mathscr{G}}}_{r},where s′≔(s1,s2,0)s^{\prime} := \left({s}_{1},{s}_{2},0). Since HGrνGr=ΔGrs{{\mathscr{H}}}_{{{\mathscr{G}}}_{r}}{\nu }_{{{\mathscr{G}}}_{r}}={\Delta }_{{{\mathscr{G}}}_{r}}sfor s∈Grs\in {{\mathscr{G}}}_{r}and ∫Grp0νGr⋅ζdGr=∫Frp0divζds=0{\int }_{{{\mathscr{G}}}_{r}}{p}_{0}{\nu }_{{{\mathscr{G}}}_{r}}\cdot \zeta {\rm{d}}{{\mathscr{G}}}_{r}={\int }_{{{\mathscr{F}}}_{r}}{p}_{0}{\rm{div}}\hspace{0.33em}\zeta {\rm{d}}s=0, integration by parts and the divergence theorem imply (5.12)I[r]=∫Grω22∣s′∣2νGr⋅ζdGr=ω22∫Frdiv(∣s′∣2ζ)dGr=ω2∫Frζ⋅s′ds.I\left[r]=\mathop{\int }\limits_{{{\mathscr{G}}}_{r}}\frac{{\omega }^{2}}{2}| s^{\prime} {| }^{2}{\nu }_{{{\mathscr{G}}}_{r}}\cdot \zeta {\rm{d}}{{\mathscr{G}}}_{r}=\frac{{\omega }^{2}}{2}\mathop{\int }\limits_{{{\mathscr{F}}}_{r}}{\rm{div}}\hspace{0.33em}\left(| s^{\prime} {| }^{2}\zeta ){\rm{d}}{{\mathscr{G}}}_{r}={\omega }^{2}\mathop{\int }\limits_{{{\mathscr{F}}}_{r}}\zeta \cdot s^{\prime} {\rm{d}}s.We next calculate the first variation of I(r)I\left(r). By using the well known formulas δ0HGr=ΔGr+(HG2−2KG)r,δ0∣s′∣2=2νG⋅y′r,{\delta }_{0}{{\mathscr{H}}}_{{{\mathscr{G}}}_{r}}={\Delta }_{{\mathscr{G}}}r+\left({{\mathscr{H}}}_{{\mathscr{G}}}^{2}-2{{\mathscr{K}}}_{{\mathscr{G}}})r,\hspace{1.0em}{\delta }_{0}| s^{\prime} {| }^{2}=2{\nu }_{{\mathscr{G}}}\cdot y^{\prime} r,it follows from (5.11) that δ0I[r]=−∫GℬGrνG⋅ζdG=−∫GC^GrνG⋅ζdG.{\delta }_{0}I\left[r]=-\mathop{\int }\limits_{{\mathscr{G}}}{{\mathcal{ {\mathcal B} }}}_{{\mathscr{G}}}r{\nu }_{{\mathscr{G}}}\cdot \zeta {\rm{d}}{\mathscr{G}}=-\mathop{\int }\limits_{{\mathscr{G}}}{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}r{\nu }_{{\mathscr{G}}}\cdot \zeta {\rm{d}}{\mathscr{G}}.Conversely, the first variation of (5.12) gives δ0I[r]=ω2∫Grζ⋅y′dG{\delta }_{0}I\left[r]={\omega }^{2}\mathop{\int }\limits_{{\mathscr{G}}}r\zeta \cdot y^{\prime} {\rm{d}}{\mathscr{G}}due to a formula δ0∫Frfds=∫GfrdG{\delta }_{0}{\int }_{{{\mathscr{F}}}_{r}}f{\rm{d}}s={\int }_{{\mathscr{G}}}fr{\rm{d}}{\mathscr{G}}. Combining the aforementioned equalities, we obtain the required equality.□We next give the proof of Lemma 5.5.Proof of Lemma 5.5First, we show that the closed subspace X˜0{\widetilde{X}}_{0}of X0{X}_{0}is e−Aqt{e}^{-{{\mathcal{A}}}_{q}t}-invariant, i.e., e−AqtX˜0⊂X˜0{e}^{-{{\mathcal{A}}}_{q}t}{\widetilde{X}}_{0}\subset {\widetilde{X}}_{0}for any t≥0t\ge 0. According to the classical semigroup theory (cf. [20, Thm. 4.5.1]), it suffices to show that there is a real number c∗{c}_{\ast }such that for every λ>c∗\lambda \gt {c}_{\ast }, the space X˜0{\widetilde{X}}_{0}is an invariant subspace of R(λ;−Aq)R\left(\lambda ;\hspace{0.33em}-{{\mathcal{A}}}_{q}), the resolvent of −Aq-{{\mathcal{A}}}_{q}. To this end, we shall consider the following resolvent problem (5.13)λU^∗−Lω,yU^∗+∇Q^∗=F˜,in F,divU^∗=0,in F,PG(2μD(U^∗)νG)=0,on G,2μD(U^∗)νG⋅νG−Q^∗+C^Gh^∗=0,on G,λh^∗−(P0GU^∗)⋅νG=G˜,on G\left\{\begin{array}{ll}\lambda {\widehat{U}}_{\ast }-{L}_{\omega ,y}{\widehat{U}}_{\ast }+\nabla {\widehat{Q}}_{\ast }=\widetilde{F},& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{\widehat{U}}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({\widehat{U}}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-{\widehat{Q}}_{\ast }+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=\widetilde{G},& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.for given (F˜,G˜)∈X˜0\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}. Here, λ∈Σε,λ∗\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{\ast }}is a resolvent parameter, where ε∈(0,π/2)\varepsilon \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2)and λ∗≥max(λ4,3∣ω∣){\lambda }_{\ast }\ge \max \left({\lambda }_{4},3| \omega | )are constants. From Lemma 5.4, the resolvent set ρ(−Aq)\rho \left(-{{\mathcal{A}}}_{q})of −Aq-{{\mathcal{A}}}_{q}contains Σε,λ∗{\Sigma }_{\varepsilon ,{\lambda }_{\ast }}.Integrating (5.13)5{}_{5}, for m=1,2,3,4m=1,2,3,4, it holds 0=(G˜,φm)G=(λh^∗−(P0GU^∗)⋅νG,φm)G=λ(h^∗,φm)G−∫Fdiv((P0GU^∗)φm)dy.0={\left(\widetilde{G},{\varphi }_{m})}_{{\mathscr{G}}}={\left(\lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}},{\varphi }_{m})}_{{\mathscr{G}}}=\lambda {\left({\widehat{h}}_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}-\mathop{\int }\limits_{{\mathscr{F}}}{\rm{div}}\hspace{0.33em}\left(\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast }){\varphi }_{m}){\rm{d}}y.In the following, we write U^∗≔(U^∗(1),U^∗(2),U^∗(3)){\widehat{U}}_{\ast }:= \left({\widehat{U}}_{\ast }^{\left(1)},{\widehat{U}}_{\ast }^{\left(2)},{\widehat{U}}_{\ast }^{\left(3)}). As div(P0GU^∗)=divU^∗=0{\rm{div}}\hspace{0.33em}\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })={\rm{div}}\hspace{0.33em}{\widehat{U}}_{\ast }=0in F{\mathscr{F}}and ∂ℓφm{\partial }_{\ell }{\varphi }_{m}, ℓ=1,2,3\ell =1,2,3, are constants, we observe ∫Fdiv((P0GU^∗)φm)dy=∫F(div(P0GU^∗))φmdy+∑ℓ=13∫FU^∗(ℓ)−1∣F∣∫FU^∗(ℓ)dy(∂ℓφm)dy=0.\mathop{\int }\limits_{{\mathscr{F}}}{\rm{div}}\hspace{0.33em}\left(\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast }){\varphi }_{m}){\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}\left({\rm{div}}\hspace{0.33em}\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })){\varphi }_{m}{\rm{d}}y+\mathop{\sum }\limits_{\ell =1}^{3}\mathop{\int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(\ell )}-\frac{1}{| {\mathscr{F}}| }\mathop{\int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(\ell )}{\rm{d}}y\right)\left({\partial }_{\ell }{\varphi }_{m}){\rm{d}}y=0.Since λ≠0\lambda \ne 0, this gives (h^∗,φm)G=0{\left({\widehat{h}}_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}=0for m=1,2,3,4m=1,2,3,4.Employing the similar argument as mentioned earlier, we can show U^∗{\widehat{U}}_{\ast }satisfies the orthogonal conditions. In fact, integrating (5.13)1{}_{1}, for ℓ=1,2,3\ell =1,2,3, we have 0=(F˜,eℓ)F=(λU^∗−μΔU^∗+2ω(e3×U^∗)+∇Q^∗,eℓ)F=λ(U^∗,eℓ)F+2ω(e3×U^∗,eℓ)F+(C^Gh^∗,νG⋅eℓ)G.0={\left(\widetilde{F},{e}_{\ell })}_{{\mathscr{F}}}={\left(\lambda {\widehat{U}}_{\ast }-\mu \Delta {\widehat{U}}_{\ast }+2\omega \left({e}_{3}\times {\widehat{U}}_{\ast })+\nabla {\widehat{Q}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}=\lambda {\left({\widehat{U}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}+2\omega {\left({e}_{3}\times {\widehat{U}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}+{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot {e}_{\ell })}_{{\mathscr{G}}}.By Proposition 5.6, it holds (C^Gh^∗,νG⋅eα)G=−ω2∫Gh^∗yαdG,(α=1,2),(C^Gh^∗,νG⋅e3)G=0,{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot {e}_{\alpha })}_{{\mathscr{G}}}=-{\omega }^{2}\mathop{\int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{\alpha }{\rm{d}}{\mathscr{G}},\hspace{1.0em}\left(\alpha =1,2),\hspace{1.0em}{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot {e}_{3})}_{{\mathscr{G}}}=0,and thus, we obtain λ∫FU^∗(1)dy−2ω∫FU^∗(2)dy−ω2∫Gh^∗y1dG=0,λ∫FU^∗(2)dy+2ω∫FU^∗(1)dy−ω2∫Gh^∗y2dG=0,λ∫FU^∗(3)dy=0.\begin{array}{rcl}\lambda \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(1)}{\rm{d}}y-2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(2)}{\rm{d}}y-{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{\rm{d}}{\mathscr{G}}& =& 0,\\ \lambda \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(2)}{\rm{d}}y+2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(1)}{\rm{d}}y-{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{\rm{d}}{\mathscr{G}}& =& 0,\\ \lambda \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(3)}{\rm{d}}y& =& 0.\end{array}Recalling that h^∗{\widehat{h}}_{\ast }satisfies (h^∗,φm)G=0{\left({\widehat{h}}_{\ast },{\varphi }_{m})}_{{\mathscr{G}}}=0for m=1,2,3,4m=1,2,3,4, we obtain (U^∗,eℓ)F=0{\left({\widehat{U}}_{\ast },{e}_{\ell })}_{{\mathscr{F}}}=0, ℓ=1,2,3\ell =1,2,3, due to λ≠0,±2iω\lambda \ne 0,\pm 2i\omega .We next multiply the equation (5.13)1{}_{1}by eℓ×y{e}_{\ell }\times y, ℓ=1,2,3\ell =1,2,3, and integrate over F{\mathscr{F}}, which gives (F˜,eℓ×y)F=λ(U^∗,eℓ×y)F−2ωδℓ,3∫FU^∗⋅y′dy+2ω(e3×U^∗,eℓ×y)F+(C^Gh^∗,νG⋅(eℓ×y))G.{\left(\widetilde{F},{e}_{\ell }\times y)}_{{\mathscr{F}}}=\lambda {\left({\widehat{U}}_{\ast },{e}_{\ell }\times y)}_{{\mathscr{F}}}-2\omega {\delta }_{\ell ,3}\left(\mathop{\int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }\cdot y^{\prime} {\rm{d}}y\right)+2\omega {\left({e}_{3}\times {\widehat{U}}_{\ast },{e}_{\ell }\times y)}_{{\mathscr{F}}}+{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{\ell }\times y))}_{{\mathscr{G}}}.Here, δℓ,3{\delta }_{\ell ,3}stands for the Kronecker delta. Since Proposition 5.6 gives (C^Gh^∗,νG⋅(e1×y))G=ω2∫Gh^∗y2y3dG,(C^Gh^∗,νG⋅(e2×y))G=−ω2∫Gh^∗y1y3dG,(C^Gh^∗,νG⋅(e3×y))G=0,\begin{array}{rcl}{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{1}\times y))}_{{\mathscr{G}}}& =& {\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{2}\times y))}_{{\mathscr{G}}}& =& -{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widehat{h}}_{\ast },{\nu }_{{\mathscr{G}}}\cdot \left({e}_{3}\times y))}_{{\mathscr{G}}}& =& 0,\end{array}it holds (F˜,e1×y)F=λ(U^∗,e1×y)F−2ω∫FU^∗(1)y3dy+ω2∫Gh^∗y2y3dG,(F˜,e2×y)F=λ(U^∗,e2×y)F−2ω∫FU^∗(2)y3dy−ω2∫Gh^∗y1y3dG,(F˜,e3×y)F=λ(U^∗,e3×y)F.\begin{array}{rcl}{\left(\widetilde{F},{e}_{1}\times y)}_{{\mathscr{F}}}& =& \lambda {\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(1)}{y}_{3}{\rm{d}}y+{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left(\widetilde{F},{e}_{2}\times y)}_{{\mathscr{F}}}& =& \lambda {\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-2\omega \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\widehat{U}}_{\ast }^{\left(2)}{y}_{3}{\rm{d}}y-{\omega }^{2}\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}},\\ {\left(\widetilde{F},{e}_{3}\times y)}_{{\mathscr{F}}}& =& \lambda {\left({\widehat{U}}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}.\end{array}Conversely, we have ω∫GG˜yαy3dG=ω∫G(λh^∗−(P0GU^∗)⋅νG)yαy3dG=λω∫Gh^∗yαy3dG−ω∫F(U^∗(α)y3+U^∗(3)yα)dy\omega \mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=\omega \mathop{\int }\limits_{{\mathscr{G}}}(\lambda {\widehat{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widehat{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=\lambda \omega \mathop{\int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}-\omega \mathop{\int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(\alpha )}{y}_{3}+{\widehat{U}}_{\ast }^{\left(3)}{y}_{\alpha }){\rm{d}}ywith α=1,2\alpha =1,2. Since F˜\widetilde{F}and G˜\widetilde{G}satisfy the conditions: ∫FF˜⋅(eα×y)dy=ω∫GG˜yαy3dG,(α=1,2),∫FF˜⋅(e3×y)dy=0,\mathop{\int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{\alpha }\times y){\rm{d}}y=\omega \mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},\hspace{1.0em}\left(\alpha =1,2),\hspace{1.0em}\mathop{\int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{3}\times y){\rm{d}}y=0,we observe (5.14)λ(U^∗,e1×y)F−ω∫Gh^∗y1y3dG−ω∫F(U^∗(1)y3−U^∗(3)y1)dy−ω∫Gh^∗y2y3dG=0,λ(U^∗,e2×y)F−ω∫Gh^∗y2y3dG+ω∫F(U^∗(3)y2−U^∗(2)y3)dy−ω∫Gh^∗y1y3dG=0,λ(U^∗,e3×y)F=0.\begin{array}{l}\lambda \left({\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)-\omega \left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(1)}{y}_{3}-{\widehat{U}}_{\ast }^{\left(3)}{y}_{1}){\rm{d}}y-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)=0,\\ \lambda \left({\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)+\omega \left(\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\left({\widehat{U}}_{\ast }^{\left(3)}{y}_{2}-{\widehat{U}}_{\ast }^{\left(2)}{y}_{3}){\rm{d}}y-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)=0,\\ \lambda {\left({\widehat{U}}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}=0.\end{array}Noting that U^∗(1)y3−U^∗(3)y1=U^∗⋅(e2×y),U^∗(2)y3−U^∗(3)y2=U^∗⋅(e1×y),{\widehat{U}}_{\ast }^{\left(1)}{y}_{3}-{\widehat{U}}_{\ast }^{\left(3)}{y}_{1}={\widehat{U}}_{\ast }\cdot \left({e}_{2}\times y),\hspace{1.0em}{\widehat{U}}_{\ast }^{\left(2)}{y}_{3}-{\widehat{U}}_{\ast }^{\left(3)}{y}_{2}={\widehat{U}}_{\ast }\cdot \left({e}_{1}\times y),it follows from (5.14)1,2{\left(5.14)}_{1,2}that λ(U^∗,e1×y)F−ω∫Gh^∗y1y3dG−ω(U^∗,e2×y)F−ω∫Gh^∗y2y3dG=0,λ(U^∗,e2×y)F−ω∫Gh^∗y2y3dG+ω(U^∗,e1×y)F−ω∫Gh^∗y1y3dG=0.\begin{array}{rcl}\lambda \left({\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)-\omega \left({\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)& =& 0,\\ \lambda \left({\left({\widehat{U}}_{\ast },{e}_{2}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{2}{y}_{3}{\rm{d}}{\mathscr{G}}\right)+\omega \left({\left({\widehat{U}}_{\ast },{e}_{1}\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{1}{y}_{3}{\rm{d}}{\mathscr{G}}\right)& =& 0.\end{array}Hence, by λ≠0,±iω\lambda \ne 0,\pm i\omega , we arrive at (U^∗,eα×y)F−ω∫Gh^∗yαy3dG=0,(α=1,2),(U^∗,e3×y)F=0,{\left({\widehat{U}}_{\ast },{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}{\widehat{h}}_{\ast }{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=0,\hspace{1.0em}\left(\alpha =1,2),\hspace{1.0em}{\left({\widehat{U}}_{\ast },{e}_{3}\times y)}_{{\mathscr{F}}}=0,which implies (U^∗,h^∗)∈D(A˜q)\left({\widehat{U}}_{\ast },{\widehat{h}}_{\ast })\in {\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q}). Accordingly, we have the R(λ;−Aq)R\left(\lambda ;-{{\mathcal{A}}}_{q})-invariance of X˜0{\widetilde{X}}_{0}for any λ∈Σε,λ∗\lambda \in {\Sigma }_{\varepsilon ,{\lambda }_{\ast }}, which implies Σε,λ∗⊂ρ(−A˜q){\Sigma }_{\varepsilon ,{\lambda }_{\ast }}\subset \rho \left(-{\widetilde{{\mathcal{A}}}}_{q})(cf. [10, Prop. A.2.8]). Hence, the C0{C}_{0}-semigroup {e−Aqt∣X˜0}t≥0{\left\{{e}^{-{{\mathcal{A}}}_{q}t}{| }_{{\widetilde{X}}_{0}}\right\}}_{t\ge 0}is indeed analytic.□In the following, we denote the induced C0{C}_{0}-semigroup by {e−A˜qt}t≥0≔{e−Aqt∣X˜0}t≥0.\{{e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}{\}}_{t\ge 0}:= \{{e}^{-{{\mathcal{A}}}_{q}t}{| }_{{\widetilde{X}}_{0}}{\}}_{t\ge 0}.As F{\mathscr{F}}and G{\mathscr{G}}are compact, we can prove that 0∈ρ(A˜q)0\in \rho \left({\widetilde{{\mathcal{A}}}}_{q}), which immediately implies the exponential stability of the analytic C0{C}_{0}-semigroup generated by −A˜q-{\widetilde{{\mathcal{A}}}}_{q}.Theorem 5.7Let 1<q<∞1\lt q\lt \infty . For any (F˜,G˜)∈X˜0\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}and t>0t\gt 0, there exist positive constants C and β∗{\beta }_{\ast }such that∣e−A˜qt(F˜,G˜)∣X0≤Ce−β∗t∣(F˜,G˜)∣X0| {e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}\left(\widetilde{F},\widetilde{G}){| }_{{X}_{0}}\le C{e}^{-{\beta }_{\ast }t}| \left(\widetilde{F},\widetilde{G}){| }_{{X}_{0}}is valid, i.e., {e−A˜qt}t≥0{\left\{{e}^{-{\widetilde{{\mathcal{A}}}}_{q}t}\right\}}_{t\ge 0}is exponentially stable on X˜0{\widetilde{X}}_{0}.ProofFrom Lemma 5.5, there exists λ∗>0{\lambda }_{\ast }\gt 0such that Σε,λ∗⊂ρ(−A˜q){\Sigma }_{\varepsilon ,{\lambda }_{\ast }}\subset \rho \left(-{\widetilde{{\mathcal{A}}}}_{q})for ε∈(0,π/2)\varepsilon \in \left(0,\pi \hspace{0.1em}\text{/}\hspace{0.1em}2). It remains to prove out result for λ∈Qλ∗≔{λ∈C:Reλ≥0,∣λ∣≤λ∗}\lambda \in {Q}_{{\lambda }_{\ast }}:= \left\{\lambda \in {\mathbb{C}}\hspace{0.33em}:\hspace{0.33em}{\rm{Re}}\hspace{0.33em}\lambda \ge 0,| \lambda | \le {\lambda }_{\ast }\right\}. We define R∗≔R(2λ∗;−A˜q):X˜0→X1∩X˜0⊂X˜0.{R}_{\ast }:= R\left(2{\lambda }_{\ast };-{\widetilde{{\mathcal{A}}}}_{q})\hspace{0.33em}:\hspace{0.33em}{\widetilde{X}}_{0}\to {X}_{1}\cap {\widetilde{X}}_{0}\subset {\widetilde{X}}_{0}.Since F{\mathscr{F}}and G{\mathscr{G}}are compact, it follows from the Rellich theorem that R∗{R}_{\ast }is a compact operator from X˜0{\widetilde{X}}_{0}into itself. For any λ∈Qλ∗\lambda \in {Q}_{{\lambda }_{\ast }}, rewriting I+A˜qI+{\widetilde{{\mathcal{A}}}}_{q}by (λI+A˜q)(F˜,G˜)=(I+(λ−2λ∗)R∗)(2λ∗I+Aq)(F˜,G˜)\left(\lambda I+{\widetilde{{\mathcal{A}}}}_{q})\left(\widetilde{F},\widetilde{G})=\left(I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast })\left(2{\lambda }_{\ast }I+{{\mathcal{A}}}_{q})\left(\widetilde{F},\widetilde{G})for (F˜,G˜)∈X˜0\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}, we observe that Qλ∗⊂ρ(−A˜q){Q}_{{\lambda }_{\ast }}\subset \rho \left(-{\widetilde{{\mathcal{A}}}}_{q})follows from the Fredholm alternative theorem and the injection of I+(λ−2λ∗)R∗I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast }. To see this, for any λ∈Qλ∗\lambda \in {Q}_{{\lambda }_{\ast }}, take (F˜,G˜)∈Ker(I+(λ−2λ∗)R∗)⊂X˜0\left(\widetilde{F},\widetilde{G})\in {\rm{Ker}}\hspace{0.33em}\left(I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast })\subset {\widetilde{X}}_{0}, i.e., (I+(λ−2λ∗)R∗)(F˜,G˜)=0for any(F˜,G˜)∈X˜0.\left(I+\left(\lambda -2{\lambda }_{\ast }){R}_{\ast })\left(\widetilde{F},\widetilde{G})=0\hspace{1.0em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}\left(\widetilde{F},\widetilde{G})\in {\widetilde{X}}_{0}.By the definition of R∗{R}_{\ast }, we see that (F˜,G˜)\left(\widetilde{F},\widetilde{G})belongs to D(A˜q){\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q})and satisfies (5.15)(λ+A˜q)(F˜,G˜)=0for anyλ∈Qλ∗.\left(\lambda +{\widetilde{{\mathcal{A}}}}_{q})\left(\widetilde{F},\widetilde{G})=0\hspace{1.0em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}\lambda \in {Q}_{{\lambda }_{\ast }}.Notice that the equation (5.15) is equivalent to (5.16)λF˜−Lω,yF˜+∇K(F˜,G˜)=0,in F,PG(2μD(F˜)νG)=0,on G,2μD(F˜)νG⋅νG−K(F˜,G˜)+C^GG˜=0,on G,λG˜−(P0GF˜)⋅νG=0,on G\left\{\begin{array}{ll}\lambda \widetilde{F}-{L}_{\omega ,y}\widetilde{F}+\nabla K\left(\widetilde{F},\widetilde{G})=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left(\widetilde{F}){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left(\widetilde{F}){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-K\left(\widetilde{F},\widetilde{G})+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ \lambda \widetilde{G}-\left({P}_{0}^{{\mathscr{G}}}\widetilde{F})\cdot {\nu }_{{\mathscr{G}}}=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}\end{array}\right.with the orthogonal conditions (5.17)∫FF˜dy=∫FF˜⋅(e3×y)dy=0,∫FF˜⋅(eα×y)dy=ω∫GG˜yαy3dG,(α=1,2),∫GG˜dG=∫GG˜yℓdG=0,(ℓ=1,2,3).\left\{\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\widetilde{F}{\rm{d}}y& =& \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{3}\times y){\rm{d}}y=0,\\ \mathop{\displaystyle \int }\limits_{{\mathscr{F}}}\widetilde{F}\cdot \left({e}_{\alpha }\times y){\rm{d}}y& =& \omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}},& \left(\alpha =1,2),\\ \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}{\rm{d}}{\mathscr{G}}& =& \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\ell }{\rm{d}}{\mathscr{G}}=0,& \left(\ell =1,2,3).\end{array}\right.In the following, we may assume (F˜,G˜)∈D(A˜2)\left(\widetilde{F},\widetilde{G})\in {\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{2}). In fact, the boundedness of F{\mathscr{F}}implies D(A˜q)↪D(A˜2){\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{q})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{2})for 2≤q<∞2\le q\lt \infty . Besides, when 1<q≤21\lt q\le 2, by the bootstrap argument and Sobolev embedding theorem, we see (F˜,G˜)∈D(A˜2)\left(\widetilde{F},\widetilde{G})\in {\mathsf{D}}\left({\widetilde{{\mathcal{A}}}}_{2})as well. Using the divergence theorem and (5.17), it follows from (5.16) that (5.18)0=λ∣F˜∣L2(F)2+2μ∣D(F˜)∣L2(F)2+(C^GG˜,F˜⋅νG)G=λ∣F˜∣L2(F)2+2μ∣D(F˜)∣L2(F)2+λ¯(C^GG˜,G˜)G=λ∣F˜∣L2(F)2+2μ∣D(F˜)∣L2(F)2+λ¯ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2,\begin{array}{rcl}0& =& \lambda | \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+2\mu | D\left(\widetilde{F}){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G},\widetilde{F}\cdot {\nu }_{{\mathscr{G}}})}_{{\mathscr{G}}}\\ & =& \lambda | \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+2\mu | D\left(\widetilde{F}){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\overline{\lambda }{\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G},\widetilde{G})}_{{\mathscr{G}}}\\ & =& \lambda | \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+2\mu | D\left(\widetilde{F}){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\overline{\lambda }\left\{{\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\right\},\end{array}where ΨG{\Psi }_{{\mathscr{G}}}is the quadratic form given in (1.12). Now, we decompose F˜=F˜⊥+∑α=1,2dα[G˜](eα×y)\widetilde{F}={\widetilde{F}}^{\perp }+\sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]\left({e}_{\alpha }\times y)with F˜⊥{\widetilde{F}}^{\perp }satisfying ∫FF˜⊥dy=∫FF˜⊥⋅(eℓ×y)dy=0,(ℓ=1,2,3),\mathop{\int }\limits_{{\mathscr{F}}}{\widetilde{F}}^{\perp }{\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}{\widetilde{F}}^{\perp }\cdot \left({e}_{\ell }\times y){\rm{d}}y=0,\hspace{1.0em}\left(\ell =1,2,3),and ∣F˜∣L2(F)2=∣F˜⊥∣L2(F)2+∑α=1,2dα[G˜]2Sα.| \widetilde{F}{| }_{{L}^{2}\left({\mathscr{F}})}^{2}=| {\widetilde{F}}^{\perp }{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\sum _{\alpha =1,2}{d}_{\alpha }{\left[\widetilde{G}]}^{2}{{\mathcal{S}}}_{\alpha }.Then, (5.18) becomes 0=λ∣F˜⊥∣L2(F)2+∑α=1,2dα[G˜]2Sα+2μ∣D(F˜⊥)∣L2(F)2+λ¯ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2.0=\lambda \left(| {\widetilde{F}}^{\perp }{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\sum _{\alpha =1,2}{d}_{\alpha }{\left[\widetilde{G}]}^{2}{{\mathcal{S}}}_{\alpha }\right)+2\mu | D\left({\widetilde{F}}^{\perp }){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\overline{\lambda }\left\{{\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\right\}.Taking the real part yields 0=2μ∣D(F˜⊥)∣L2(F)2+(Reλ)∣F˜⊥∣L2(F)2+∑α=1,2dα[G˜]2Sα+ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2.0=2\mu | D\left({\widetilde{F}}^{\perp }){| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\left({\rm{Re}}\hspace{0.33em}\lambda )\left\{| {\widetilde{F}}^{\perp }{| }_{{L}^{2}\left({\mathscr{F}})}^{2}+\sum _{\alpha =1,2}{d}_{\alpha }{\left[\widetilde{G}]}^{2}{{\mathcal{S}}}_{\alpha }+{\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\right\}.According to Assumption 1.1, we see that Reλ≥0{\rm{Re}}\hspace{0.33em}\lambda \ge 0implies D(F˜⊥)=0D\left({\widetilde{F}}^{\perp })=0in F{\mathscr{F}}. Hence, it follows from the second Korn inequality that F˜⊥=0{\widetilde{F}}^{\perp }=0in F{\mathscr{F}}. Then, the first equation of (5.16) takes the form λ∑α=1,2dα[G˜](eα×y)−Lω,y∑α=1,2dα[G˜](eα×y)+∇K(F˜,G˜)=0in F.\lambda \sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]\left({e}_{\alpha }\times y)-{L}_{\omega ,y}\left(\sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]\left({e}_{\alpha }\times y)\right)+\nabla K\left(\widetilde{F},\widetilde{G})=0\hspace{1.0em}\hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{}.Taking the curl{\rm{curl}}in this equation leads to λ∑α=1,2dα[G˜]eα−ω(d2[G˜]e1−d1[G˜]e2)=0.\lambda \sum _{\alpha =1,2}{d}_{\alpha }\left[\widetilde{G}]{e}_{\alpha }-\omega ({d}_{2}\left[\widetilde{G}]{e}_{1}-{d}_{1}\left[\widetilde{G}]{e}_{2})=0.Hence, we obtain λd1[G˜]=ωd2[G˜]\lambda {d}_{1}\left[\widetilde{G}]=\omega {d}_{2}\left[\widetilde{G}]and λd2[G˜]=−ωd1[G˜]\lambda {d}_{2}\left[\widetilde{G}]=-\omega {d}_{1}\left[\widetilde{G}]. Besides, it follows from (5.17)2{}_{2}that λdα[G˜]=−λωSα∫GG˜yαy3dG=−λSα∫F∑β=1,2dβ[G˜](eβ×y)⋅(eα×y)dy,(α=1,2).\lambda {d}_{\alpha }\left[\widetilde{G}]=-\lambda \frac{\omega }{{{\mathcal{S}}}_{\alpha }}\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}{y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}=-\frac{\lambda }{{{\mathcal{S}}}_{\alpha }}\mathop{\int }\limits_{{\mathscr{F}}}\sum _{\beta =1,2}{d}_{\beta }\left[\widetilde{G}]\left({e}_{\beta }\times y)\cdot \left({e}_{\alpha }\times y){\rm{d}}y,\hspace{1.0em}\left(\alpha =1,2).Setting S˜α=∫F(yα2−y32)dy,(α=1,2),{\widetilde{{\mathcal{S}}}}_{\alpha }=\mathop{\int }\limits_{{\mathscr{F}}}({y}_{\alpha }^{2}-{y}_{3}^{2}){\rm{d}}y,\hspace{1.0em}\left(\alpha =1,2),we easily observe λd1[G˜]=−ωS˜1S1d2[G˜]=−λS˜1S1d1[G˜],λd2[G˜]=ωS˜2S2d1[G˜]=−λS˜2S2d2[G˜],\lambda {d}_{1}\left[\widetilde{G}]=-\frac{\omega {\widetilde{{\mathcal{S}}}}_{1}}{{{\mathcal{S}}}_{1}}{d}_{2}\left[\widetilde{G}]=-\lambda \frac{{\widetilde{{\mathcal{S}}}}_{1}}{{{\mathcal{S}}}_{1}}{d}_{1}\left[\widetilde{G}],\hspace{1.0em}\lambda {d}_{2}\left[\widetilde{G}]=\frac{\omega {\widetilde{{\mathcal{S}}}}_{2}}{{{\mathcal{S}}}_{2}}{d}_{1}\left[\widetilde{G}]=-\lambda \frac{{\widetilde{{\mathcal{S}}}}_{2}}{{{\mathcal{S}}}_{2}}{d}_{2}\left[\widetilde{G}],i.e., d1[G˜]=d2[G˜]=0{d}_{1}\left[\widetilde{G}]={d}_{2}\left[\widetilde{G}]=0, as follows: S˜α+Sα=∫F((yα2−y32)+(∣y∣2−yα2))dy=∫F∣y′∣2dy≠0.{\widetilde{{\mathcal{S}}}}_{\alpha }+{{\mathcal{S}}}_{\alpha }=\mathop{\int }\limits_{{\mathscr{F}}}(({y}_{\alpha }^{2}-{y}_{3}^{2})+\left(| y{| }^{2}-{y}_{\alpha }^{2})){\rm{d}}y=\mathop{\int }\limits_{{\mathscr{F}}}| y^{\prime} {| }^{2}{\rm{d}}y\ne 0.Thus, we arrive at F˜=0\widetilde{F}=0in F{\mathscr{F}}, which, combined with (5.16)1{}_{1}, implies that the pressure term K(F˜,G˜)K\left(\widetilde{F},\widetilde{G})is equal to some constant p0{{\mathsf{p}}}_{0}. From (5.16)2{}_{2}, we have C^GG˜=p0{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}={{\mathsf{p}}}_{0}on G{\mathscr{G}}, but it follows from ∫GC^GG˜dG=0{\int }_{{\mathscr{G}}}{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}{\rm{d}}{\mathscr{G}}=0that p0=0{{\mathsf{p}}}_{0}=0, i.e., C^GG˜=0{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G}=0on G{\mathscr{G}}. Hence, we observe that 0=(C^GG˜,G˜)G=ΨG(G˜,G˜)+ω2S3∫GG˜∣y′∣2dG2≥c∣G˜∣L2(G)2+ω2S3∫GG˜∣y′∣2dG2,0={\left({\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}\widetilde{G},\widetilde{G})}_{{\mathscr{G}}}={\Psi }_{{\mathscr{G}}}\left(\widetilde{G},\widetilde{G})+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2}\ge c| \widetilde{G}{| }_{{L}^{2}\left({\mathscr{G}})}^{2}+\frac{{\omega }^{2}}{{{\mathcal{S}}}_{3}}{\left(\mathop{\int }\limits_{{\mathscr{G}}}\widetilde{G}| y^{\prime} {| }^{2}{\rm{d}}{\mathscr{G}}\right)}^{2},which implies G˜=0\widetilde{G}=0on G{\mathscr{G}}. Therefore, there are no eigenvalues λ∈Qλ∗\lambda \in {Q}_{{\lambda }_{\ast }}of −A˜2-{\widetilde{{\mathcal{A}}}}_{2}. This completes the proof.□By using Theorem 5.7, we complete the decay estimate of solutions to (5.3).Theorem 5.8Assume that 1<p,q<∞1\lt p,q\lt \infty , 1/p<δ≤11\hspace{0.1em}\text{/}\hspace{0.1em}p\lt \delta \le 1, and 1/p+1/(2q)≠δ−1/21\hspace{0.1em}\text{/}p+1\text{/}\hspace{0.1em}\left(2q)\ne \delta -1\hspace{0.1em}\text{/}\hspace{0.1em}2. Let v∗{v}_{\ast }and η∗{\eta }_{\ast }be functions obtained in Theorem 5.2. Then, the solution (u∗,h∗)\left({u}_{\ast },{h}_{\ast })to (5.3) enjoys the estimate∣eε0t(u∗,h∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({u}_{\ast },{h}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).with a constant C independent of T.ProofBy the variation of constants formula, we see that (U∗,h∗)(⋅,t)=∫0te−A˜q(t−s)(2λ2f˜w(⋅,s),2λ2η˜∗(⋅,s))ds\left({U}_{\ast },{h}_{\ast })\left(\cdot ,t)=\underset{0}{\overset{t}{\int }}{e}^{-{\widetilde{{\mathcal{A}}}}_{q}\left(t-s)}(2{\lambda }_{2}{\widetilde{f}}_{w}\left(\cdot ,s),2{\lambda }_{2}{\widetilde{\eta }}_{\ast }\left(\cdot ,s)){\rm{d}}sare solutions to (5.7) with Q∗=K(U∗,h∗){Q}_{\ast }=K\left({U}_{\ast },{h}_{\ast }). Then, Theorem 5.7 yields ∣(U∗,h∗)(⋅,t)∣X0≤Cλ2∫0te−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0ds≤Cλ2∫0te−β∗(t−s)ds1/p′∫0te−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0pds1/p≤Cλ2β∗−1/p′∫0te−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0pds1/p.\begin{array}{rcl}| \left({U}_{\ast },{h}_{\ast })\left(\cdot ,t){| }_{{X}_{0}}& \le & C{\lambda }_{2}\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}{\rm{d}}s\\ & \le & C{\lambda }_{2}{\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}{\rm{d}}s\right)}^{1\text{/}p^{\prime} }{\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & \le & C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }{\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}^{p}{\rm{d}}s\right)}^{1\text{/}p}.\end{array}Hence, for every 0<ε0<β∗/p0\lt {\varepsilon }_{0}\lt {\beta }_{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}p, it holds (5.19)∫0T(eε0t∣(U∗,h∗)(⋅,t)∣X0)pdt≤Cλ2β∗−1/p′∫0T∫0teε0tpe−β∗(t−s)∣(f˜w,η˜∗)(⋅,s)∣X0pdsdt=Cλ2β∗−1/p′∫0T∫0te−(β∗−ε0p)(t−s)(eε0s∣(f˜w,η˜∗)(⋅,s)∣X0)pdsdt=Cλ2β∗−1/p′∫0T(eε0s∣(f˜w,η˜∗)(⋅,s)∣X0)p∫sTe−(β∗−ε0p)(t−s)dtds=Cλ2β∗−1/p′(β∗−ε0p)−1∫0T(eε0s∣(f˜w,η˜∗)(⋅,s)∣X0)pds.\begin{array}{rcl}\underset{0}{\overset{T}{\displaystyle \int }}{({e}^{{\varepsilon }_{0}t}| \left({U}_{\ast },{h}_{\ast })\left(\cdot ,t){| }_{{X}_{0}})}^{p}{\rm{d}}t& \le & C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }\underset{0}{\overset{T}{\displaystyle \int }}\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{{\varepsilon }_{0}tp}{e}^{-{\beta }_{\ast }\left(t-s)}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}^{p}{\rm{d}}s\right){\rm{d}}t\\ & =& C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }\underset{0}{\overset{T}{\displaystyle \int }}\left(\underset{0}{\overset{t}{\displaystyle \int }}{e}^{-\left({\beta }_{\ast }-{\varepsilon }_{0}p)\left(t-s)}{({e}^{{\varepsilon }_{0}s}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}})}^{p}{\rm{d}}s\right){\rm{d}}t\\ & =& C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }\underset{0}{\overset{T}{\displaystyle \int }}{({e}^{{\varepsilon }_{0}s}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}})}^{p}\left(\underset{s}{\overset{T}{\displaystyle \int }}{e}^{-\left({\beta }_{\ast }-{\varepsilon }_{0}p)\left(t-s)}{\rm{d}}t\right){\rm{d}}s\\ & =& C{\lambda }_{2}{\beta }_{\ast }^{-1\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} }{\left({\beta }_{\ast }-{\varepsilon }_{0}p)}^{-1}\underset{0}{\overset{T}{\displaystyle \int }}{({e}^{{\varepsilon }_{0}s}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}})}^{p}{\rm{d}}s.\end{array}Besides, we have ∣(f˜w,η˜∗)(⋅,s)∣X0≤∣f˜w∣Lq(F)+∣η˜∗∣Bq,q2−1/q(G)≤C(∣v˜∗∣Lq(F)+(1+ω)∣η˜∗∣Bq,q2−1/q(G))≤C(∣v∗∣Lq(F)+(1+ω)∣η∗∣Bq,q2−1/q(G))\begin{array}{rcl}| ({\widetilde{f}}_{w},{\widetilde{\eta }}_{\ast })\left(\cdot ,s){| }_{{X}_{0}}& \le & | {\widetilde{f}}_{w}{| }_{{L}^{q}\left({\mathscr{F}})}+| {\widetilde{\eta }}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})}\\ & \le & C(| {\widetilde{v}}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+\left(1+\omega )| {\widetilde{\eta }}_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})})\\ & \le & C(| {v}_{\ast }{| }_{{L}^{q}\left({\mathscr{F}})}+\left(1+\omega )| {\eta }_{\ast }{| }_{{B}_{q,q}^{2-1\text{/}q}\left({\mathscr{G}})})\end{array}for any s∈(0,T)s\in \left(0,T). Hence, it follows from the estimate (5.19) and Corollary 5.3 that (5.20)∣eε0t(U∗,h∗)(⋅,t)∣Lp(J;X0)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F))| {e}^{{\varepsilon }_{0}t}\left({U}_{\ast },{h}_{\ast })\left(\cdot ,t){| }_{{L}^{p}\left(J;{X}_{0})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})for t∈(0,T)t\in \left(0,T)and ε0∈(0,β∗/p){\varepsilon }_{0}\in \left(0,{\beta }_{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}p), where CCis independent of TTand tt. Besides, by (5.7)5{\left(5.7)}_{5}, we also obtain the estimate (5.21)∣eε0th∗(⋅,t)∣Fp,q,δ1−1/(2q)(J;Lq(G))≤C∣eε0th∗(⋅,t)∣Hδ1,p(J;Lq(F))∩Lδp(J;H2,q(F))≤C(∣eε0t∂sh∗(⋅,t)∣Lδp(J;Lq(F))+∣eε0th∗(⋅,t)∣Lδp(J;H2,q(F)))≤C(∣eε0t(U∗,η˜∗)(⋅,t)∣Lδp(J;Lq(F))+∣eε0th∗(⋅,t)∣Lδp(J;H2,q(F)))≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F))\begin{array}{rcl}| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{F}_{p,q,\delta }^{1-1\text{/}\left(2q)}\left(J;{L}^{q}\left({\mathscr{G}}))}& \le & C| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{H}_{\delta }^{1,p}\left(J;{L}^{q}\left({\mathscr{F}}))\cap {L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}}))}\\ & \le & C(| {e}^{{\varepsilon }_{0}t}{\partial }_{s}{h}_{\ast }\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}}))})\\ & \le & C(| {e}^{{\varepsilon }_{0}t}\left({U}_{\ast },{\widetilde{\eta }}_{\ast })\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}{h}_{\ast }\left(\cdot ,t){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}}))})\\ & \le & C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})})\end{array}with a constant CCindependent of ttand TT. If (U˜∗,h˜∗)\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast })satisfies the shifted equations ∂tU˜∗+2λ2U˜∗−Lω,yU˜∗+∇K(U˜∗,h˜∗)=2λ2(f˜w+U∗),in F,divU˜∗=0,in F,PG(2μD(U˜∗)νG)=0,on G,2μD(U˜∗)νG⋅νG−K(U˜∗,h˜∗)+C^Gh˜∗=0,on G,∂th˜∗+2λ2h˜∗−(P0GU˜∗)⋅νG=2λ2(η˜∗+h∗),on G,U˜∗(0)=0,in F,h˜∗(0)=0,on G,\left\{\begin{array}{ll}{\partial }_{t}{\widetilde{U}}_{\ast }+2{\lambda }_{2}{\widetilde{U}}_{\ast }-{L}_{\omega ,y}{\widetilde{U}}_{\ast }+\nabla K\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast })=2{\lambda }_{2}({\widetilde{f}}_{w}+{U}_{\ast }),& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\rm{div}}\hspace{0.33em}{\widetilde{U}}_{\ast }=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {{\mathcal{P}}}_{{\mathscr{G}}}\left(2\mu D\left({\widetilde{U}}_{\ast }){\nu }_{{\mathscr{G}}})=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ 2\mu D\left({\widetilde{U}}_{\ast }){\nu }_{{\mathscr{G}}}\cdot {\nu }_{{\mathscr{G}}}-K\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast })+{\widehat{{\mathcal{C}}}}_{{\mathscr{G}}}{\widetilde{h}}_{\ast }=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\partial }_{t}{\widetilde{h}}_{\ast }+2{\lambda }_{2}{\widetilde{h}}_{\ast }-\left({P}_{0}^{{\mathscr{G}}}{\widetilde{U}}_{\ast })\cdot {\nu }_{{\mathscr{G}}}=2{\lambda }_{2}\left({\widetilde{\eta }}_{\ast }+{h}_{\ast }),& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\\ {\widetilde{U}}_{\ast }\left(0)=0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}{\mathscr{F}}\text{},\\ {\widetilde{h}}_{\ast }\left(0)=0,& \hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{},\end{array}\right.we obtain from Corollary 5.3, (5.20), and (5.21) that ∣eε0t(U˜∗,h˜∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({\widetilde{U}}_{\ast },{\widetilde{h}}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).Noting that U∗{U}_{\ast }and h∗{h}_{\ast }satisfy (5.7), the uniqueness of solutions implies U˜∗=U∗{\widetilde{U}}_{\ast }={U}_{\ast }and h˜∗=h∗{\widetilde{h}}_{\ast }={h}_{\ast }for any t∈(0,T)t\in \left(0,T), and hence, we observe ∣eε0t(U∗,h∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({U}_{\ast },{h}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).Recalling the decomposition (5.6), we arrive at ∣eε0t(u∗,h∗)∣E1,δ(J;F)×E4,δ(J;G)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}\left({u}_{\ast },{h}_{\ast }){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).This completes the proof of Theorem 5.8.□Step 3: The completion of the proof of Theorem 5.1. Finally, let us derive the estimates of (u,q,h)\left(u,q,h). Recalling (5.4) and (5.5), it follows from Theorems 5.2 and 5.8 that ∣eε0t∂t(u,h)∣Lδp(J;X0)≤C(∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)).| {e}^{{\varepsilon }_{0}t}{\partial }_{t}\left(u,h){| }_{{L}_{\delta }^{p}\left(J;{X}_{0})}\le C(| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}).Besides, we also have ∣eε0t(u,h)∣Lδp(J;X1)≤∣eε0t(u,h)∣Lδp(J;Lq(F)×Lq(G))+∑j=1,2∣eε0t∇ju∣Lδp(J;Lq(F))+∑k=1,2,3∣eε0t∇kh∣Lδp(J;Lq(G))≤C[∣eε0t(v∗,η∗)∣Lδp(J;H2,q(F)×Bq,q3−1/q(G))+∣eε0t(u∗,h∗)∣Lδp(J;H2,q(F)×Bq,q3−1/q(G))+∫0T(eε0s∣(u∗(⋅,s),1)F∣)pds1/p+∑α=1,2∫0T(eε0s∣(u∗(⋅,s),eα×y)F−ω∫Gh∗(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u∗(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h∗(⋅,s),φm)G∣)pds1/p≤C[∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)+∫0T(eε0s∣(u∗(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u∗(⋅,s),eα×y)F−ω∫Gh∗(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u∗(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h∗(⋅,s),φm)G∣)pds1/p,\begin{array}{rcl}| {e}^{{\varepsilon }_{0}t}\left(u,h){| }_{{L}_{\delta }^{p}\left(J;{X}_{1})}& \le & | {e}^{{\varepsilon }_{0}t}\left(u,h){| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}})\times {L}^{q}\left({\mathscr{G}}))}+\displaystyle \sum _{j=1,2}| {e}^{{\varepsilon }_{0}t}{\nabla }^{j}u{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+\displaystyle \sum _{k=1,2,3}| {e}^{{\varepsilon }_{0}t}{\nabla }^{k}h{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{G}}))}\\ & \le & C{[}| {e}^{{\varepsilon }_{0}t}\left({v}_{\ast },{\eta }_{\ast }){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}}))}+| {e}^{{\varepsilon }_{0}t}\left({u}_{\ast },{h}_{\ast }){| }_{{L}_{\delta }^{p}\left(J;{H}^{2,q}\left({\mathscr{F}})\times {B}_{q,q}^{3-1\text{/}q}\left({\mathscr{G}}))}\\ & & +{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}\left(\underset{0}{\overset{T}{\displaystyle \int }}({e}^{{\varepsilon }_{0}s}\phantom{\rule[-7pt]{}{0ex}}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}\right.\\ & & {\left.{\left.\left.-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & & \left.+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({h}_{\ast }\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right]\le C{[}| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}\\ & & +| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & & +\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left({u}_{\ast }\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ & & \left.+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({h}_{\ast }\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right],\end{array}where CCis independent of TT. Finally, we estimate eε0th{e}^{{\varepsilon }_{0}t}hin the Fp,q,δ2−/q(J;Lq(G)){F}_{p,q,\delta }^{2-\hspace{0.1em}\text{/}\hspace{0.1em}q}\left(J;\hspace{0.33em}{L}^{q}\left({\mathscr{G}}))-norm. It holds ∣eε0th∣Fp,q,δ2−1/q(J;Lq(G))≤C∣eε0th∣Hδ2,p(J;Lq(F))∩Hδ1,p(J;H2,q(F))≤C(∣eε0t∂t2h∣Lδp(J;Lq(F))+∣eε0th∣Hδ1,p(J;H2,q(F))).| {e}^{{\varepsilon }_{0}t}h{| }_{{F}_{p,q,\delta }^{2-1\text{/}q}\left(J;{L}^{q}\left({\mathscr{G}}))}\le C| {e}^{{\varepsilon }_{0}t}h{| }_{{H}_{\delta }^{2,p}\left(J;{L}^{q}\left({\mathscr{F}}))\cap {H}_{\delta }^{1,p}\left(J;{H}^{2,q}\left({\mathscr{F}}))}\le C(| {e}^{{\varepsilon }_{0}t}{\partial }_{t}^{2}h{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}h{| }_{{H}_{\delta }^{1,p}\left(J;{H}^{2,q}\left({\mathscr{F}}))}).From (4.1)5{\left(4.1)}_{5}, we observe that ∂t2h−(P0G∂tu)⋅νG=∂tfhon G.{\partial }_{t}^{2}h-\left({P}_{0}^{{\mathscr{G}}}{\partial }_{t}u)\cdot {\nu }_{{\mathscr{G}}}={\partial }_{t}{f}_{h}\hspace{1.0em}\hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\mathscr{G}}\text{}.Hence, we obtain ∣eε0t∂t2h∣Lδp(J;Lq(F))≤C(∣eε0t(P0G∂tu)⋅νG∣Lδp(J;Lq(F))+∣eε0t∂tfh∣Lδp(J;Lq(F)))≤C(∣eε0tu∣E1,δ(J;F)+∣eε0tfh∣F4,δ(J;G)),\begin{array}{rcl}| {e}^{{\varepsilon }_{0}t}{\partial }_{t}^{2}h{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}& \le & C(| {e}^{{\varepsilon }_{0}t}\left({P}_{0}^{{\mathscr{G}}}{\partial }_{t}u)\cdot {\nu }_{{\mathscr{G}}}{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))}+| {e}^{{\varepsilon }_{0}t}{\partial }_{t}{f}_{h}{| }_{{L}_{\delta }^{p}\left(J;{L}^{q}\left({\mathscr{F}}))})\\ & \le & C(| {e}^{{\varepsilon }_{0}t}u{| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})}+| {e}^{{\varepsilon }_{0}t}{f}_{h}{| }_{{{\mathbb{F}}}_{4,\delta }\left(J;{\mathscr{G}})}),\end{array}which implies ∣eε0th∣Fp,q,δ2−1/q(J;Lq(G))≤C(∣eε0t(u,h)∣E1,δ(J;F)×E4,δ(J;G)+∣eε0tfh∣F4,δ(J;G)).| {e}^{{\varepsilon }_{0}t}h{| }_{{F}_{p,q,\delta }^{2-1\text{/}q}\left(J;{L}^{q}\left({\mathscr{G}}))}\le C(| {e}^{{\varepsilon }_{0}t}\left(u,h){| }_{{{\mathbb{E}}}_{1,\delta }\left(J;{\mathscr{F}})\times {{\mathbb{E}}}_{4,\delta }\left(J;{\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}{f}_{h}{| }_{{{\mathbb{F}}}_{4,\delta }\left(J;{\mathscr{G}})}).It follows from (5.4) and (5.5) that ∫0T(eε0s∣(u∗(⋅,s),1)F∣)pds1/p+∫0T(eε0s∣(u∗(⋅,s),e3×y)F∣)pds1/p+∑α=1,2∫0Teε0s(u∗(⋅,s),eα×y)F−ω∫Gh∗(⋅,s)yαy3dGpds1/p≤C∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F)},∑m=14∫0T(eε0s∣(h∗(⋅,s),φm)G∣)pds1/p≤C∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p+∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(G)+∣eε0t(fu,gd,guτ,guv,fh)∣Fδ(J;F),\begin{array}{l}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({u}_{\ast }\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left({u}_{\ast }\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}{h}_{\ast }\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\le C\left\{{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right.\\ \hspace{1.0em}\hspace{1.0em}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}+| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}\},\\ \mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left({h}_{\ast }\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\le C\left\{\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}({f}_{u},{g}_{d},{g}_{u\tau },{g}_{uv},{f}_{h}){| }_{{{\mathbb{F}}}_{\delta }\left(J;{\mathscr{F}})}\right\},\end{array}where CCdepends on ω\omega but is independent of TT. This completes the proof of Theorem 5.1.6The nonlinear problem6.1Local existenceWe now show the local existence result for given initial data (v0,Γ0)∈Bq,p2(δ−1/p)(Ω(0))3×Bq,p2+δ−1/p−1/q(G),\left({v}_{0},{\Gamma }_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left(\Omega \left(0))}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}}),which are subject to the compatibility conditions the compatibility conditions divv0=0in Ω(0),PΓ0[μ(∇v0+[∇v0]⊤)]=0on Γ0{\rm{div}}\hspace{0.33em}{v}_{0}=0\hspace{1.0em}\hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\Omega \left(0)\text{},\hspace{1.0em}{{\mathcal{P}}}_{{\Gamma }_{0}}\left[\mu \left(\nabla {v}_{0}+{\left[\nabla {v}_{0}]}^{\top })]=0\hspace{1.0em}\hspace{0.1em}\text{on\hspace{0.5em}}\hspace{0.1em}{\Gamma }_{0}\text{}and condition (1.4). We define the nonlinear mapping N=(N1,N2,N3,N4,N5,N6)N=\left({N}_{1},{N}_{2},{N}_{3},{N}_{4},{N}_{5},{N}_{6})by N1≔Fu(u,q,h)N2≔Gd(u,h)=divGdiv(u,h),N3≔Guτ(u,h)N4≔Guv(u,h)+G0(h)N5≔Fh(u,h)+F(u,h)\begin{array}{rcl}{N}_{1}& := & {F}_{u}\left(u,q,h)\hspace{1.0em}{N}_{2}:= {G}_{d}\left(u,h)={\rm{div}}\hspace{0.33em}{G}_{{\rm{div}}}\left(u,h),\\ {N}_{3}& := & {G}_{u\tau }\left(u,h)\hspace{1.0em}{N}_{4}:= {G}_{uv}\left(u,h)+{G}_{0}\left(h)\hspace{1.0em}{N}_{5}:= {F}_{h}\left(u,h)+F\left(u,h)\end{array}respectively. We set UT≔{z=(u,q,TrG[q],h)∈Eδ(J;F):∣h∣L∞(G×J)<ε}{{\mathbb{U}}}_{T}:= \left\{z=\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\hspace{0.33em}:\hspace{0.33em}| h{| }_{{L}^{\infty }\left({\mathscr{G}}\times J)}\lt \varepsilon \right\}with J=(0,T)J=\left(0,T), T>0T\gt 0, and 0<ε<10\lt \varepsilon \lt 1. Then we have the following proposition.Proposition 6.1Let (p,q,δ)\left(p,q,\delta )satisfy (1.13). Then it holds(1)NNis a real analytic mapping from UT{{\mathbb{U}}}_{T}to Fδ(J;F){{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})and N(0)=DN(0)=0N\left(0)=DN\left(0)=0.(2)DN(z)∈ℒ(UT,Fδ(J;F))DN\left(z)\in {\mathcal{ {\mathcal L} }}\left({{\mathbb{U}}}_{T},{{\mathbb{F}}}_{\delta }\left(J;\hspace{0.33em}{\mathscr{F}})).Here, DNDNrepresents the Fréchet derivative of NN.ProofSince (p,q,δ)\left(p,q,\delta )satisfies (1.13), we have the following assertions, see [39, Lem. 5.3]: (i)E1,δ(J;F)↪BUC1(J;BUC(F)){{\mathbb{E}}}_{1,\delta }\left(J;\hspace{0.33em}{\mathscr{F}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{1}\left(J;\hspace{0.33em}{\rm{BUC}}\left({\mathscr{F}})).(ii)E3,δ(J;G)↪BUC(J;BUC(G)){{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{\rm{BUC}}\left(J;\hspace{0.33em}{\rm{BUC}}\left({\mathscr{G}})).(iii)E4,δ(J;G)↪BUC1(J;BUC1(G))∩BUC(J;BUC2(G)){{\mathbb{E}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\rm{BUC}}}^{1}\left(J;\hspace{0.33em}{{\rm{BUC}}}^{1}\left({\mathscr{G}}))\cap {\rm{BUC}}\left(J;\hspace{0.33em}{{\rm{BUC}}}^{2}\left({\mathscr{G}})).(iv)E3,δ(J;G){{\mathbb{E}}}_{3,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})and F4,δ(J;G){{\mathbb{F}}}_{4,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})are multiplication algebras.Here, in assertions (i)–(iii), the embedding constants are independent of T>0T\gt 0if the time traces vanish at t=0t=0. The polynomial structure of the nonlinearity NNwith respect to uu, qq, and hhgives mapping properties for NN. This completes the proof.□Using the aforementioned proposition, we can obtain the local existence of classical solution to (1.1). Since the proof is standard (cf. [39, Sec. 5]), we may omit the detail.Theorem 6.2Let T>0T\gt 0be a given constant. Assume conditions (1.13) hold. Then there exists a constant ε=ε(T)>0\varepsilon =\varepsilon \left(T)\gt 0such that for arbitrary initial data (u0,η0)∈Bq,p2(δ−1/p)(F)3×Bq,p2+δ−1/p−1/q(G)\left({u}_{0},{\eta }_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfying the compatibility conditions(6.1)divu0=Gd(u0,h0)=divGdiv(u0,h0),inF,PF[μ(∇u0+[∇u0]⊤)]=Guτ(u0,h0),onF,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{\rm{div}}\hspace{0.33em}{u}_{0}={G}_{d}\left({u}_{0},{h}_{0})={\rm{div}}\hspace{0.33em}{G}_{{\rm{div}}}\left({u}_{0},{h}_{0}),& {in}\hspace{0.33em}{\mathscr{F}},\\ {{\mathcal{P}}}_{{\mathscr{F}}}\left[\mu \left(\nabla {u}_{0}+{\left[\nabla {u}_{0}]}^{\top })]={G}_{u\tau }\left({u}_{0},{h}_{0}),& {on}\hspace{0.33em}{\mathscr{F}},\end{array}\right.and the smallness condition∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(F)≤ε,| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{F}})}\le \varepsilon ,the transformed problem (3.7) has a unique solution (u,q,TrG[q],η)∈Eδ((0,T);F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta )\in {{\mathbb{E}}}_{\delta }\left(\left(0,T);\hspace{0.33em}{\mathscr{F}}). Furthermore, the solution is indeed real analytic in F×(0,T){\mathscr{F}}\times \left(0,T)and, especially, ℳ≔⋃t∈(0,T)(Γ(t)×{t}){\mathcal{ {\mathcal M} }}:= {\bigcup }_{t\in \left(0,T)}\left(\Gamma \left(t)\times \left\{t\right\})is a real analytic manifold.Remark 6.3To remove the smallness condition on the initial velocity field v0{v}_{0}, one has to consider the modified term Fh(u,h)+b⋅∇Gh{F}_{h}\left(u,h)+b\cdot {\nabla }_{{\mathscr{G}}}hinstead of Fh(u,h){F}_{h}\left(u,h), where b∈F4,δ(J;G)3b\in {{\mathbb{F}}}_{4,\delta }{\left(J;{\mathscr{G}})}^{3}is taken such that b(0)=TrG[PGu0]b\left(0)={{\rm{Tr}}}_{{\mathscr{G}}}\left[{{\mathcal{P}}}_{{\mathscr{G}}}{u}_{0}]. In fact, Fh(u,h){F}_{h}\left(u,h)cannot be small in the norm of F5,δ(J;G){{\mathbb{F}}}_{5,\delta }\left(J;\hspace{0.33em}{\mathscr{G}})even if ∇G∣h∣L∞(G){\nabla }_{{\mathscr{G}}}| h{| }_{{L}^{\infty }\left({\mathscr{G}})}is small. However, to make our discussion simple, we keep the smallness assumption on v0{v}_{0}. Notice that the local existence result for arbitrary large initial velocity already obtained by Shibata [24], and see also [27, Thm. 3.6.1].6.2Global existence and convergenceFinally, we prove Theorem 1.2. In the following, we suppose that the initial data (u0,η0)∈Bq,p2(δ−1/p)(F)3×Bq,p2+δ−1/p−1/q(G)\left({u}_{0},{\eta }_{0})\in {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}{\left({\mathscr{F}})}^{3}\times {B}_{q,p}^{2+\delta -1\hspace{0.1em}\text{/}p-1\text{/}\hspace{0.1em}q}\left({\mathscr{G}})satisfies the smallness condition: (6.2)∣u0∣Bq,p2(δ−1/p)(F)+∣h0∣Bq,p2+δ−1/p−1/q(F)≤ε| {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}+| {h}_{0}{| }_{{B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{F}})}\le \varepsilon with some small ε>0\varepsilon \gt 0as well as the compatibility conditions (6.1). Since we will choose ε\varepsilon small eventually, we may suppose 0<ε<10\lt \varepsilon \lt 1. By Theorem 6.2, for given T0>0{T}_{0}\gt 0, there exists ε∗>0{\varepsilon }_{\ast }\gt 0such that (3.7) admits a unique solution (u,q,TrG[q],h)∈Eδ((0,T0);F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left(\left(0,{T}_{0});\hspace{0.33em}{\mathscr{F}}). In the following, we may assume ε<ε∗\varepsilon \lt {\varepsilon }_{\ast }. We further assume that the system (3.7) admits a solution (u,q,TrG[q],h)∈Eδ(J0;F)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)\in {{\mathbb{E}}}_{\delta }\left({J}_{0};\hspace{0.33em}{\mathscr{F}})with J0=(0,T0){J}_{0}=\left(0,{T}_{0}). We shall show that the solution (u,q,TrG[q],h)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)can be prolong to the time interval R+{{\mathbb{R}}}_{+}. To this end, it suffices to verify the a priori estimate (6.3)∣eε0t(u,q,TrG[q],h)∣Eδ(0,T;F)≤C(∣(u0,η0)∣Bq,p2(δ−1/p)(F)×Bq,p2+δ−1/p−1/q(G)+∣eε0t(u,q,TrG[q],h)∣Eδ(0,T;F)2)| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;{\mathscr{F}})}\le C(| \left({u}_{0},{\eta }_{0}){| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})\times {B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2})for any T∈(0,T0]T\in \left(0,{T}_{0}], where a constant CCis independent of ε\varepsilon , TT, and T0{T}_{0}. Here, ε0{\varepsilon }_{0}is the same constant as in Theorem 5.1. In fact, combining the local existence result and the a priori estimate (6.3), a standard bootstrap argument implies the desired result.From Theorem 5.1 and Proposition 6.1, we easily find that (u,q,TrG[q],h)\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],h)enjoys the estimate ∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)≤C[∣(u0,η0)∣Bq,p2(δ−1/p)(F)×Bq,p2+δ−1/p−1/q(G)+∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2+∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p.\begin{array}{l}| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;{\mathscr{F}})}\\ \hspace{1.0em}\le C{[}| \left({u}_{0},{\eta }_{0}){| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})\times {B}_{q,p}^{2+\delta -1\text{/}p-1\text{/}q}\left({\mathscr{G}})}+| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2}\\ \hspace{2.0em}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}\left.+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\right].\end{array}A similar argument given in [26, Sec. 6] gives (6.4)∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p+∫0T(eε0s∣(u(⋅,s),e3×y)F∣)pds1/p+∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p≤C∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2.\begin{array}{l}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\displaystyle \sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\displaystyle \int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\hspace{1.0em}+{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),{e}_{3}\times y)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\mathop{\displaystyle \sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\displaystyle \int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\\ \hspace{1.0em}\le C| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2}.\end{array}In fact, by [32, Sec. 2], it follows 0=∫Ω˜(t)dz−∫Fdy=∫Gh−h22HG+h33KGdG,0=∫Ω˜(t)zℓdz−∫Fyℓdy=∫Gh−h22HG+h33KGyℓ+νG(ℓ)h22−h33HG+h44KGdG,\begin{array}{rcl}0& =& \mathop{\displaystyle \int }\limits_{\widetilde{\Omega }\left(t)}{\rm{d}}z-\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{\rm{d}}y=\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\left(h-\frac{{h}^{2}}{2}{{\mathscr{H}}}_{{\mathscr{G}}}+\frac{{h}^{3}}{3}{{\mathscr{K}}}_{{\mathscr{G}}}\right){\rm{d}}{\mathscr{G}},\\ 0& =& \mathop{\displaystyle \int }\limits_{\widetilde{\Omega }\left(t)}{z}_{\ell }{\rm{d}}z-\mathop{\displaystyle \int }\limits_{{\mathscr{F}}}{y}_{\ell }{\rm{d}}y=\mathop{\displaystyle \int }\limits_{{\mathscr{G}}}\left[\left(h-\frac{{h}^{2}}{2}{{\mathscr{H}}}_{{\mathscr{G}}}+\frac{{h}^{3}}{3}{{\mathscr{K}}}_{{\mathscr{G}}}\right){y}_{\ell }+{\nu }_{{\mathscr{G}}}^{\left(\ell )}\left(\frac{{h}^{2}}{2}-\frac{{h}^{3}}{3}{{\mathscr{H}}}_{{\mathscr{G}}}+\frac{{h}^{4}}{4}{{\mathscr{K}}}_{{\mathscr{G}}}\right)\right]{\rm{d}}{\mathscr{G}},\end{array}which gives the estimate (6.5)∑m=14∫0T(eε0s∣(h(⋅,s),φm)G∣)pds1/p≤C∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2,\mathop{\sum }\limits_{m=1}^{4}{\left(\underset{0}{\overset{T}{\int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(h\left(\cdot ,s),{\varphi }_{m})}_{{\mathscr{G}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}\le C| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2},cf., [26, Sec. 6]. Next, it follows from (1.4)2{\left(1.4)}_{2}and (1.6) that (v,eℓ×x)Ω(t)=(e3×y,eℓ×y)F=ωδℓ,3∫F∣y′∣2dy=−γδℓ,3,ℓ=1,2,3.{\left(v,{e}_{\ell }\times x)}_{\Omega \left(t)}={\left({e}_{3}\times y,{e}_{\ell }\times y)}_{{\mathscr{F}}}=\omega {\delta }_{\ell ,3}\mathop{\int }\limits_{{\mathscr{F}}}| y^{\prime} {| }^{2}{\rm{d}}y=-\gamma {\delta }_{\ell ,3},\hspace{1.0em}\ell =1,2,3.Hence, according to the transform explained in Section 3, we see that (6.6)(V˜,eℓ×z)Ω˜(t)+ω(e3×z,eℓ×z)Ω˜(t)=−γδℓ,3,ℓ=1,2,3.{\left(\widetilde{V},{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}+\omega {\left({e}_{3}\times z,{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}=-\gamma {\delta }_{\ell ,3},\hspace{1.0em}\ell =1,2,3.As J(h){\mathsf{J}}\left(h)describes the Jacobian of the transform Ξh{\Xi }_{h}, which is introduced in Section 3, we have (V˜,eℓ×z)Ω˜(t)=∫Fu⋅(eℓ×Ξh)J(h)dy=(u,eℓ×y)F+(u,eℓ×ξh)F+J0(h)(u,eℓ×Ξh)F{\left(\widetilde{V},{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}=\mathop{\int }\limits_{{\mathscr{F}}}u\cdot \left({e}_{\ell }\times {\Xi }_{h}){\mathsf{J}}\left(h){\rm{d}}y={\left(u,{e}_{\ell }\times y)}_{{\mathscr{F}}}+{\left(u,{e}_{\ell }\times {\xi }_{h})}_{{\mathscr{F}}}+{{\mathsf{J}}}_{0}\left(h){\left(u,{e}_{\ell }\times {\Xi }_{h})}_{{\mathscr{F}}}with ℓ=1,2,3\ell =1,2,3. Following [32, Sect. 2], for y∈Gy\in {\mathscr{G}}, we introduce J^(y,h)\widehat{{\mathsf{J}}}(y,h)defined by J^(y,h)≔∑i,j=13νG(i)(y)νG(j)(y)J(h)(I−[M1(h)]⊤)i,j,\widehat{{\mathsf{J}}}(y,h):= \mathop{\sum }\limits_{i,j=1}^{3}{\nu }_{{\mathscr{G}}}^{\left(i)}(y){\nu }_{{\mathscr{G}}}^{\left(j)}(y){\mathsf{J}}\left(h){(I-{\left[{M}_{1}\left(h)]}^{\top })}_{i,j},where M1{M}_{1}is the matrix given in Section 3. Then, by [32, p. 1772], for ℓ=1,2,3\ell =1,2,3, we may write (e3×z,eℓ×z)Ω˜(t)=(e3×y,eℓ×y)F+∫01∫G((e3×Ξrh)⋅(eℓ×Ξrh))hJ^(y,rh)dGdr,{\left({e}_{3}\times z,{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}={\left({e}_{3}\times y,{e}_{\ell }\times y)}_{{\mathscr{F}}}+\underset{0}{\overset{1}{\int }}\left(\mathop{\int }\limits_{{\mathscr{G}}}(\left({e}_{3}\times {\Xi }_{rh})\cdot \left({e}_{\ell }\times {\Xi }_{rh}))h\widehat{{\mathsf{J}}}(y,rh){\rm{d}}{\mathscr{G}}\right){\rm{d}}r,which yields the expression ω(e3×z,eℓ×z)Ω˜(t)=−γδℓ,3−ω∫Ghyℓy3dG+ωN˜ℓ(y,h).\omega {\left({e}_{3}\times z,{e}_{\ell }\times z)}_{\widetilde{\Omega }\left(t)}=-\gamma {\delta }_{\ell ,3}-\omega \mathop{\int }\limits_{{\mathscr{G}}}h{y}_{\ell }{y}_{3}{\rm{d}}{\mathscr{G}}+\omega {\widetilde{N}}_{\ell }(y,h).Here, N˜ℓ(y,h){\widetilde{N}}_{\ell }(y,h)is a nonlinear term that is given by N˜ℓ(y,h)=∫Ghyℓy3dG+∫01∫G((e3×Ξrh)⋅(eℓ×Ξrh))hJ^(y,rh)dGdr.{\widetilde{N}}_{\ell }(y,h)=\mathop{\int }\limits_{{\mathscr{G}}}h{y}_{\ell }{y}_{3}{\rm{d}}{\mathscr{G}}+\underset{0}{\overset{1}{\int }}\left(\mathop{\int }\limits_{{\mathscr{G}}}(\left({e}_{3}\times {\Xi }_{rh})\cdot \left({e}_{\ell }\times {\Xi }_{rh}))h\widehat{{\mathsf{J}}}(y,rh){\rm{d}}{\mathscr{G}}\right){\rm{d}}r.Hence, (6.6) turns into (u,eℓ×y)F−ω∫Ghyℓy3dG=−(u,eℓ×ξh)F−J0(h)(u,eℓ×Ξh)F−ωN˜ℓ(y,h),{\left(u,{e}_{\ell }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}h{y}_{\ell }{y}_{3}{\rm{d}}{\mathscr{G}}=-{\left(u,{e}_{\ell }\times {\xi }_{h})}_{{\mathscr{F}}}-{{\mathsf{J}}}_{0}\left(h){\left(u,{e}_{\ell }\times {\Xi }_{h})}_{{\mathscr{F}}}-\omega {\widetilde{N}}_{\ell }(y,h),where the right-hand side is nonlinear. Thanks to [39, Lem. 5.3], we have the estimate (6.7)∫0T(eε0s∣(u(⋅,s),1)F∣)pds1/p+∑α=1,2∫0Teε0s(u(⋅,s),eα×y)F−ω∫Gh(⋅,s)yαy3dGpds1/p≤C∣eε0t(u,q,TrG[q],η)∣Eδ(0,T;F)2.{\left(\underset{0}{\overset{T}{\int }}{\left({e}^{{\varepsilon }_{0}s}| {\left(u\left(\cdot ,s),1)}_{{\mathscr{F}}}| )}^{p}{\rm{d}}s\right)}^{1\text{/}p}+\sum _{\alpha =1,2}{\left(\underset{0}{\overset{T}{\int }}{\left({e}^{{\varepsilon }_{0}s}\left|{\left(u\left(\cdot ,s),{e}_{\alpha }\times y)}_{{\mathscr{F}}}-\omega \mathop{\int }\limits_{{\mathscr{G}}}h\left(\cdot ,s){y}_{\alpha }{y}_{3}{\rm{d}}{\mathscr{G}}\right|\right)}^{p}{\rm{d}}s\right)}^{1\text{/}p}\le C| {e}^{{\varepsilon }_{0}t}\left(u,q,{{\rm{Tr}}}_{{\mathscr{G}}}\left[q],\eta ){| }_{{{\mathbb{E}}}_{\delta }\left(0,T;\hspace{0.33em}{\mathscr{F}})}^{2}.Combining (6.5) and (6.7), we observe (6.4). Thus, we obtain (6.3).Finally, we deal with the original problem (1.1). Notice that the compatibility condition (1.14) is valid if and only if (6.1) is satisfied. Define Ξh0(y)≔y+ξh0(y){\Xi }_{{h}_{0}}(y):= y+{\xi }_{{h}_{0}}(y)with replacing hhby h0{h}_{0}in the definition of ξh{\xi }_{h}. Then, the mapping Ξh0{\Xi }_{{h}_{0}}defines a C2{C}^{2}-diffeomorphism from F{\mathscr{F}}onto Ω(0)\Omega \left(0)with inverse Ξh0−1{\Xi }_{{h}_{0}}^{-1}. This gives the existence of a constant Ch0{C}_{{h}_{0}}depending on h0{h}_{0}such that Ch0−1∣v0−v∞∣Bq,p2(δ−1/p)(Ω(0))≤∣u0∣Bq,p2(δ−1/p)(F)≤Ch0∣v0−v∞∣Bq,p2(δ−1/p)(Ω(0)).{C}_{{h}_{0}}^{-1}| {v}_{0}-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(0))}\le | {u}_{0}{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left({\mathscr{F}})}\le {C}_{{h}_{0}}| {v}_{0}-{v}_{\infty }{| }_{{B}_{q,p}^{2\left(\delta -1\text{/}p)}\left(\Omega \left(0))}.Hence, there exists ε>0\varepsilon \gt 0such that the smallness condition of Theorem 1.2 yields (6.2). Recalling the discussion in Section 3, we see that there exists a unique global classical solution (v,π,Γ)\left(v,\pi ,\Gamma )to (1.1), especially, the unique global solution (v,π,Γ)\left(v,\pi ,\Gamma )to (1.1) is real analytic. Noting Bq,p2(δ−1/p)(Ω(t))≃Bq,p2(δ−1/p)(F){B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left(\Omega \left(t))\simeq {B}_{q,p}^{2\left(\delta -1\hspace{0.1em}\text{/}\hspace{0.1em}p)}\left({\mathscr{F}}), we observe the asymptotic behavior of solutions. This completes the proof of Theorem 1.2.

Journal

Advances in Nonlinear Analysisde Gruyter

Published: Jan 1, 2023

Keywords: Navier-Stokes equations; free boundary problems; surface tension; stability; maximal regularity; Primary: 35R35; Secondary: 76D03; 76D05

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