Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur

TL;DR

This paper demonstrates that a recent relative entropy condition ensures Lopatinski stability for hyperbolic conservation laws, improving upon standard entropy conditions that can fail to guarantee stability.

Contribution

It establishes that Leger and Vasseur's entropy condition implies Lopatinski stability, offering a significant advancement over traditional entropy criteria.

Findings

Relative entropy condition implies Lopatinski stability.

Standard entropy condition can fail to ensure stability.

Leger and Vasseur's condition improves stability guarantees.

Abstract

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations