L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method

TL;DR

This paper establishes L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method, demonstrating that perturbations remain controlled over time up to a translation.

Contribution

It introduces a novel application of the relative entropy method to derive L^2 stability estimates for shock solutions in scalar conservation laws.

Findings

L^2 norm of perturbations remains bounded over time

Stability holds up to a time-dependent shock translation

Perturbations are controlled by initial data

Abstract

We consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L^2 perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L^2 norm of a perturbed solution relative to the shock wave is bounded above by the L^2 norm of the initial perturbation.