Structure of shocks in Burgers turbulence with Lévy noise initial data

Abstract

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by Lévy noise, or equivalently when the initial potential is a two-sided Lévy process $ψ_0$. When $ψ_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} ψ_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $ψ_0$ is abrupt we show that the shock structure is discrete. When $ψ_0$ is eroded we show that there are no rarefaction intervals.