The Lopatinski determinant of small shocks may vanish

TL;DR

This paper investigates the stability of small shock waves in hyperbolic systems, showing that the Lopatinski determinant, which indicates stability, can vanish even for very small non-extreme shocks under certain conditions.

Contribution

It demonstrates that the Lopatinski determinant may vanish for small non-extreme shocks associated with Metivier convex modes, challenging previous assumptions of stability.

Findings

Lopatinski determinant can vanish for small non-extreme shocks

Vanishing determinant indicates potential loss of stability

Results apply to hyperbolic conservation laws with Metivier convex modes

Abstract

The Kreiss-Majda Lopatinski determinant encodes a uniform stability property of shock wave solutions to hyperbolic systems of conservation laws in several space variables. This note deals with the Lopatinski determinant for shock waves of sufficiently small amplitude. The determinant is known to be non-zero for so-called extreme shock waves, i. e., shock waves which are asscoiated with either the slowest or the fastest mode the system displays for a given direction of propagation, if the mode is Metivier convex. The result of the note is that for arbitrarily small non-extreme shock waves associated with a Metivier convex mode, the Lopatsinki determinant may vanish.