Random Perturbation of PDEs and Fluid Dynamic Models

École d’Été de Probabilités de Saint-Flour XL – 2010

verfasst von: Franco Flandoli

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The book 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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Uniqueness and Blow-Up

Abstract

We need to keep in mind a number of examples of non-uniqueness for ODEs, first. Let us start with probably the most famous one.

Franco Flandoli

Chapter 2. Regularization by Additive Noise

Abstract

Although the final aim of these lectures is to understand the effect of noise on PDE, it is very important to investigate more and more the finite dimensional case. Even more basically, we have to understand how Brownianmotion regularizes functions, when it acts on them.

Franco Flandoli

Chapter 3. Dyadic Models

Abstract

The classical 3-dimensional Euler equations are the system of PDEs

\(\begin{array}{ll}\frac{\partial u}{\partial t} + (u \cdot \nabla) u & + \nabla p = 0\\ & {\rm div}\,\, u = 0 \end{array}\)

where \(u : [0,T]\times D \, \rightarrow \, \mathbb{R}^3\)is the velocity field of the fluid and \(p : [0,T]\times D \, \rightarrow \, \mathbb{R}^3\) is the pressure field.

Franco Flandoli

Chapter 4. Transport Equation

Abstract

In Chap. 3 on the dyadic model, we have seen that a non well posed system can be made well posed by means of a special multiplicative noise.

Franco Flandoli

Chapter 5. Other Models: Uniqueness and Singularities

Abstract

This chapter contains a number of other examples, presented for different purposes. Not only the uniqueness problem but also emergence of singularities is discussed. First, we give a few examples where noise does not change the difficulties related to these two issues; a little bit improperly, we call them “negative” examples (in spite of the fact that they are very interesting). Then we show two examples where singularities are prevented by noise: continuity equation and vortex point motion. We call them “positive” examples. The next section on nonlinear Schrödinger equation describes theoretical and numerical results both of positive and negative type. Finally, we summarize the attempts made on the 3D stochastic Navier–Stokes equations, in the direction of understanding uniqueness and singularities.

Franco Flandoli

Backmatter

Titel: Random Perturbation of PDEs and Fluid Dynamic Models
verfasst von: Franco Flandoli
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-18231-0
Print ISBN: 978-3-642-18230-3
DOI: https://doi.org/10.1007/978-3-642-18231-0