Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

TL;DR

This paper introduces a relative entropy-based theory demonstrating the uniqueness and L^2 stability of extremal shocks and contact discontinuities in conservation laws, applicable to large amplitude solutions with minimal regularity assumptions.

Contribution

It develops a general framework for stability and uniqueness of shocks and contact discontinuities using relative entropy, without requiring BV estimates or smallness conditions.

Findings

Proves L^2 stability of extremal discontinuities among bounded weak solutions.

Establishes uniqueness of solutions with a trace property.

Handles solutions with vacuum in fluid mechanics.

Abstract

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.