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SUMMARY:Uniqueness\, singularities and zero-noise limit for differential e
 quations in fluid dynamics part.3
DTSTART:20120424T101500
DTEND:20120424T120000
DTSTAMP:20260408T055727Z
UID:240df71697517f4b1c5713936baa9c51bcb42ce95dc4590a06cc86db
CATEGORIES:Conferences - Seminars
DESCRIPTION:Franco Flandoli\nThe general theme of these lectures is the re
 gularization introduced by noise in ordinary and partial differential equa
 tions. The final examples we have in mind arise from fluid dynamics\, wher
 e phenomena of non-uniqueness or blow-up appear (or it is not excluded tha
 t they may appear) in several models.\nWe start by a review of definitions
  and results of uniqueness for SDEs with nondegenerate noise and non smoot
 h drift (of Sobolev class\, Hölder continuous\, L^{p} class) with emphasi
 s on the fact that the deterministic equations with the same drift may hav
 e non-unique solutions. We also show links with the theory of uniqueness o
 f generalized Lagrangian flows. Then we present a number of examples of SP
 DEs 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 t
 he emergence of singularities. This field is more recent\, but we give exa
 mples where noise may prevent singularities which otherwise would emerge f
 or the corresponding deterministic PDE. The problem of the zero-noise lim
 it is linked to both questions of uniqueness and singularities. When a dif
 ferential equation has multiple solutions from the same initial data\, the
  zero-noise limit could be a relevant selection criterion. When singularit
 ies 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 e
 ssential to select the continuation. We discuss techniques to deal with ze
 ro-noise limits and present an example of an SPDE where we understand the 
 continuation after singularity.
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STATUS:CONFIRMED
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