Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On regular points in Burgers turbulence with stable noise initial data

Auteur(s):

Code(s) de Classification MSC:

• 35Q53 KdV-like equations (Korteweg-de Vries, Burgers, sine-Gordon, sinh-Gordon, etc.), See also {58F07}
Résumé: We study the set of regular points (i.e. the points which have not been involved into shocks up to time $t$) for the inviscid Burgers equation in dimension 1 when initial velocity is a stable Lévy noise. We prove first that when the noise is not completely asymmetric and has index $\A \in (1/2,1)$, the set of regular points is discrete a.s. and regenerative. Then, we show that in the case of the Cauchy noise, the set of regular points is uncountable, with Minkowsky dimension 0.