Short-Time Stability of Scalar Viscous Shocks in the Inviscid Limit by the Relative Entropy Method

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 by N. Leger [Arch. Ration. Mech. Anal., 199 (2011), pp. 761--778] and N. Leger and A. Vasseur [Arch. Ration. Mech. Anal., 201 (2011), pp. 271--302] to the study of an inviscid limit to a shock.