$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation

TL;DR

This paper proves that large $L^2$ perturbations of viscous shocks in multi-D scalar conservation laws decay over time without smallness restrictions, extending previous 1D results to higher dimensions.

Contribution

It establishes $L^2$ contraction and decay for large perturbations of multi-D viscous shocks without smallness assumptions, generalizing prior 1D findings.

Findings

Large $L^2$ perturbations contract over time.

Perturbations in $L^1$ decay at rate $t^{-1/4}$.

Results hold up to dynamical shift without smallness constraints.

Abstract

We consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front. However, for the transversal directions, we do not have any restrictions on flux function. We first show the contraction property for any large perturbations in $L^2$ of the planar viscous shock. If the initial $L^2$-perturbation is also in $L^1$, the large perturbation converges to zero in $L^2$ as time goes to infinity with $t^{-1/4}$ decay rate. The contraction and decay estimates hold up to dynamical shift. For the results, we do not impose any smallness conditions on the initial value. This result extends the 1D case \cite{Kang-V-1} by the first author and Vasseur to the multi-dimensional case.