$L^2$ Stability of Simple Shocks for Spatially Heterogeneous Conservation Laws

Abstract

In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable. We show that under certain assumptions, a shock wave connecting two constant states emerges in finite time for all $L^{\infty}$ initial data satisfying the same far-field conditions. Under an additional assumption on the mixed partial derivative of the flux, we establish the stability of these simple shock profiles with respect to $L^2$ perturbations. The main tools we use are Dafermos' generalised characteristics for the evolution analysis and the relative entropy method for stability.