Criteria on contractions for entropic discontinuities of systems of conservation laws

TL;DR

This paper investigates contraction properties of shocks in systems of conservation laws with convex entropy, generalizing criteria for extremal shocks and examining intermediate shocks and contact discontinuities without smallness restrictions.

Contribution

It generalizes contraction criteria for conservation law shocks, introduces necessary conditions for intermediate shocks, and applies these to magnetohydrodynamics and Euler contact discontinuities.

Findings

Generalized contraction criterion guarantees extremal shock contraction.

Intermediate shocks of magnetohydrodynamics do not satisfy contraction properties.

Contact discontinuities in Euler system have partial contraction results.

Abstract

We study the contraction properties (up to shift) for admissible Rankine-Hugoniot discontinuities of $n\times n$ systems of conservation laws endowed with a convex entropy. We first generalize the criterion developed in [47], using the spatially inhomogeneous pseudo-distance introduced in [50]. Our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems, including the Euler system. Moreover, we introduce necessary conditions for contraction, specifically targeted for intermediate shocks. As an application, we show that intermediate shocks of the two-dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties. We also investigate the contraction properties, for contact discontinuities of the Euler system, for a certain range of contraction weights. All results do not involve any smallness condition on the initial perturbation, nor on the size of the shock.