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Stochastic Analysis: A Series of LecturesA Stochastic View over the Open Problem of Well-posedness for the 3D Navier–Stokes Equations

Stochastic Analysis: A Series of Lectures: A Stochastic View over the Open Problem of... [This series of lectures discusses the open problem of well-posedness of 3D Navier–Stokes equations from the viewpoint of stochastic analysis, namely attempting to understand whether noise may improve the well-posedness. Results and obstructions of the deterministic theory are first recalled. Then the difficulties met to prove weak well-posedness by additive noise are discussed, in relation with Girsanov transform and Kolmogorov equations. Finally, the vorticity equation is considered and it is shown that a linearized version of it, under a suitable multipicative noise, has better properties than the deterministic one, in particular the blow-up due to stretching is prevented.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Stochastic Analysis: A Series of LecturesA Stochastic View over the Open Problem of Well-posedness for the 3D Navier–Stokes Equations

Part of the Progress in Probability Book Series (volume 68)
Editors: Dalang, Robert C.; Dozzi, Marco; Flandoli, Franco; Russo, Francesco
Flandoli, Franco
Stochastic Analysis: A Series of Lectures — May 15, 2015
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References (2)

Publisher
Springer Basel
Copyright
© Springer Basel 2015
ISBN
978-3-0348-0908-5
Pages
221 –246
DOI
10.1007/978-3-0348-0909-2_8
Publisher site
See Chapter on Publisher Site

Abstract

[This series of lectures discusses the open problem of well-posedness of 3D Navier–Stokes equations from the viewpoint of stochastic analysis, namely attempting to understand whether noise may improve the well-posedness. Results and obstructions of the deterministic theory are first recalled. Then the difficulties met to prove weak well-posedness by additive noise are discussed, in relation with Girsanov transform and Kolmogorov equations. Finally, the vorticity equation is considered and it is shown that a linearized version of it, under a suitable multipicative noise, has better properties than the deterministic one, in particular the blow-up due to stretching is prevented.]

Published: May 15, 2015

Keywords: Navier–Stokes equations; stochastic forcing; Kolmogorov equations; linearized vorticity equation; transport type noise; Primary: 35Q30; 60H15; secondary: 76D05; 35R60

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