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

TL;DR

This paper studies the formation and stability of shock waves in scalar conservation laws with spatially varying flux, demonstrating finite-time emergence and $L^2$ stability under certain conditions.

Contribution

It establishes $L^2$ stability of simple shocks in spatially heterogeneous conservation laws, extending understanding of shock behavior in variable flux settings.

Findings

Shock waves form in finite time for all bounded initial data with fixed far-field states.

Under specific assumptions, these shocks are stable in the $L^2$ norm.

The analysis employs Dafermos' generalized characteristics and the relative entropy method.

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.