$L^2$ 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.