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 ${\psi }_0$. When ${\psi }_0$ is abrupt in the sense of Vigon or has bounded variation with ${\mathrm{limsup}}_{|h|\downarrow 0}{h}^{-2}{\psi }_0(h)=\infty$, 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.

22 pages

References in corpus (1)

• Passage of Lévy Processes across Power Law Boundaries at Small Times

Cited by in corpus (2)

• Scalar Conservation Laws with white noise initial data
• Thin and Thick Strip Passage Times for Lévy Flights and Lévy Processes