Structure of shocks in Burgers turbulence with L\'evy noise initial data

TL;DR

This paper investigates the shock structure in Burgers turbulence with initial data modeled by Lévy noise, revealing conditions for shock discreteness, zero velocity points, and the absence of rarefaction intervals.

Contribution

It characterizes the shock structure and regular points in Burgers turbulence with Lévy noise initial data, including conditions for discreteness and regularity.

Findings

Set of zero velocity points is regenerative.

Shock structure is discrete when initial data is abrupt.

No rarefaction intervals when initial data is eroded.

Abstract

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by L\'evy noise, or equivalently when the initial potential is a two-sided L\'evy process $\psi_0$. When $\psi_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty$, 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 $\psi_0$ is abrupt we show that the shock structure is discrete. When $\psi_0$ is eroded we show that there are no rarefaction intervals.