Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux

TL;DR

This paper proves that large bounded perturbations around multi-dimensional viscous shocks decay over time, extending 1D results to higher dimensions and providing quantitative convergence rates.

Contribution

It extends contraction results for viscous shocks from 1D to multi-dimensional settings with periodic transversal directions, including decay rates for large perturbations.

Findings

Contraction property holds for large initial perturbations in multi-D.

Large perturbations decay at rate t^{-1/4} in L^2.

First quantitative convergence estimate to planar shock under large perturbations.

Abstract

We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in $L^2$ of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial $L^2$-perturbation is also in $L^1$, then the large perturbation decays of rate $t^{-1/4}$ in $L^2$, up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations.