Nonlocalised damping estimates for hyperbolic relaxation systems in one space dimensions

TL;DR

This paper introduces a new method using characteristics to derive damping estimates for hyperbolic relaxation systems, extending previous results to broader settings and enabling stability analysis under nonlocalised perturbations.

Contribution

It generalizes damping estimates from $L^2$ to $L^$ and from symmetric to non-symmetric systems, advancing the stability theory of shock profiles.

Findings

Extended damping estimates to $L^$-case.

Generalized estimates to non-symmetric systems.

Facilitated stability analysis under nonlocalised perturbations.

Abstract

In this paper, we present a new approach to obtain so-called damping estimates for self-similar solutions to general hyperbolic relaxation systems applying the method of characteristics. Such damping estimates are an important part of the stability theory of shock profiles, where they enable the closure of nonlinear stability arguments. We extend the damping estimates obtained in Mascia and Zumbrun (2005) from the $L^2$-case to the $L^\infty$-case and, at the same time, generalize the $L^2$-estimates to the non-symmetric setting. Our estimates open the door to a general stability theory of shock profiles of hyperbolic relaxation systems under nonlocalised perturbations in one space dimension.