L^2-type contraction of viscous shocks for large family of scalar conservation laws

TL;DR

This paper proves that small viscous shocks in 1D scalar conservation laws with convex flux are L2 stable regardless of dissipation strength, using a novel relative entropy method with a spatially-inhomogeneous pseudo-norm.

Contribution

It introduces a new stability proof for viscous shocks that works for large perturbations and dissipation levels, expanding understanding of shock stability.

Findings

Viscous shocks are L2 stable regardless of dissipation strength.

The relative entropy method with a spatially-inhomogeneous pseudo-norm is effective.

Stability holds for large perturbations in scalar conservation laws.

Abstract

In this paper we study small shocks of 1D scalar viscous conservation laws with uniformly convex flux and nonlinear dissipation. We show that such shocks are L2 stable independent of the strength of the dissipation, even with large perturbations. The proof uses the relative entropy method with a spatially-inhomogeneous psuedo-norm.