Random Perturbation of PDEs and Fluid Dynamic Models: École D’Été de Probabilités de Saint-Flour XL – 2010This volume deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices. |
Contents
Chapter 1 Introduction to Uniqueness and BlowUp | 1 |
Chapter 2 Regularization by Additive Noise | 17 |
Chapter 3 Dyadic Models | 70 |
Chapter 4 Transport Equation | 101 |
Chapter 5 Other Models Uniqueness and Singularities | 132 |
References | 161 |
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Random Perturbation of PDEs and Fluid Dynamic Models: École d’Été de ... Franco Flandoli Limited preview - 2011 |
Common terms and phrases
additive noise assumption blow-up bounded Brownian motions Cetraro Chap computations continuity equation converges d'Été de Probabilités Debussche defined denote deterministic Differential Equations dimension dyadic model École d'Été Editors Euler equations Example existence finite dimensional Flandoli fluid dynamics function Girsanov given heat equation hence Hölder continuous infinite initial condition integral inviscid Itô formula Kolmogorov equation Lebesgue Lebesgue measure Lecture Notes Lemma linear Lipschitz Math Mathematical multiplicative noise Navier-Stokes equations non-uniqueness nonlinear occupation measure parabolic pathwise uniqueness point vortices Prato Probabilités de Saint-Flour problem proof is complete prove random perturbations regular solutions Remark result Röckner Saint-Flour singular SPDE stochastic differential equations Stratonovich Theorem Theory transport equation uniqueness in law vector fields VU(n weak solutions x+Ws zero ди