Relative Entropy Contractions for Extremal Shocks of Nonlinear Hyperbolic Systems without Genuine Nonlinearity

TL;DR

This paper extends the theory of $a$-contraction to establish $L^2$-stability of extremal shocks in nonlinear hyperbolic systems lacking genuine nonlinearity, including applications to elastodynamics.

Contribution

It demonstrates that $a$-contraction can be used to prove stability of certain shocks in systems without genuine nonlinearity, broadening the scope of stability analysis.

Findings

$L^2$-stability up to shift for extremal shocks.

Application to 2x2 nonlinear elastodynamics.

Validity for shocks with concave-convex or convex-concave characteristic fields.

Abstract

We study extremal shocks of $1$-d hyperbolic systems of conservation laws which fail to be genuinely nonlinear. More specifically, we consider either $1$- or $n$-shocks in characteristic fields which are either concave-convex or convex-concave in the sense of LeFloch. We show that the theory of $a$-contraction can be applied to obtain $L^2$-stability up to shift for these shocks in a class of weak solutions to the conservation law whose shocks obey the Lax entropy condition. Our results apply in particular to the $2 \times 2$ system of nonlinear elastodynamics.