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

### Limits clusters in the inviscid Burgers turbulence with certain random initial velocities

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 infinite time shock limits given certain Markovian initial velocities to the inviscid Burgers turbulence. Specifically, we consider the one-sided case where initial velocities are zero on the negative half-line and follow a time-homogeneous nice Markov process $X$ on the positive half-line. Finite shock limits occur if the Markov process is transient tending to infinity. They form a Poisson point process if $X$ is spectrally negative. We give an explicit description when $X$ is furthermore spatially homogeneous (a L\'evy process) or a self-similar process on $(0,\infty)$. We also consider the two-sided case where we suppose an independent dual process in the negative spatial direction. Both spatial homogeneity and an exponential L\'evy condition lead to stationary shock limits.