Short-time stability of scalar viscous shocks in the inviscid limit by the relative entropy method

TL;DR

This paper demonstrates that scalar viscous shocks in one-dimensional conservation laws converge sharply in L^2 norm to inviscid shocks, regardless of viscous layer structure, using the relative entropy method.

Contribution

It is the first application of the relative entropy method to study the inviscid limit to shocks, providing sharp L^2 estimates for large perturbations.

Findings

Sharp L^2 convergence estimates for viscous shocks

Convergence does not depend on viscous layer structure

Applicable to large perturbations in scalar conservation laws

Abstract

We consider inviscid limits to shocks for viscous scalar conservation laws in one space dimension, with strict convex fluxes. We show that we can obtain sharp estimates in $L^2$, for a class of large perturbations and for any bounded time interval. Those perturbations can be chosen big enough to destroy the viscous layer. This shows that the fast convergence to the shock does not depend on the fine structure of the viscous layers. This is the first application of the relative entropy method developed in [22], [23] to the study of an inviscid limit to a shock.