Uniqueness, singularities and zero-noise limit for differential equations in fluid dynamics part.3
2012-04-24 10:15 - 12:00 | AAC006 http://plan.epfl.ch/?zoom=20&recenter_y=5864224.42038&recenter_x=730672.24955&layerNodes=fonds,batiments,labels,information,parkings_publics,arrets_metro&floor=0&q=AAC006
Franco Flandoli
The general theme of these lectures is the regularization introduced by noise in ordinary and partial differential equations. The final examples we have in mind arise from fluid dynamics, where phenomena of non-uniqueness or blow-up appear (or it is not excluded that they may appear) in several models.
We start by a review of definitions and results of uniqueness for SDEs with nondegenerate noise and non smooth drift (of Sobolev class, Hölder continuous, L^{p} class) with emphasis on the fact that the deterministic equations with the same drift may have non-unique solutions. We also show links with the theory of uniqueness of generalized Lagrangian flows. Then we present a number of examples of SPDEs and infinite dimensional systems where similar regularization occurs. The role of additive and bilinear multiplicative noise is discussed. For PDEs and SPDEs it is possible to investigate also the effect of noise on the emergence of singularities. This field is more recent, but we give examples where noise may prevent singularities which otherwise would emerge for the corresponding deterministic PDE. The problem of the zero-noise limit is linked to both questions of uniqueness and singularities. When a differential equation has multiple solutions from the same initial data, the zero-noise limit could be a relevant selection criterion. When singularities appear, it may happen that, due to the loss of regularity, there is no uniqueness of continuation after the singularity, so again one has a selection problem by using the zero-noise limit, but enriched by the fact that solutions have a past before the singularity time, which could be essential to select the continuation. We discuss techniques to deal with zero-noise limits and present an example of an SPDE where we understand the continuation after singularity.
