Spectral stability of weak relaxation shock profiles

TL;DR

This paper proves spectral stability of small-amplitude relaxation shocks in symmetric dissipative systems using energy estimates, extending prior results by removing a technical assumption through singular perturbation methods.

Contribution

It introduces a new approach combining Kawashima- and Goodman-type energy estimates to establish spectral stability without the noncharacteristic background equation assumption.

Findings

Spectral stability of small-amplitude relaxation shocks is established.

The method extends previous results by Plaza and Zumbrun.

The approach removes the need for the background equation to be noncharacteristic.

Abstract

Using a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun by singular perturbation techniques under an additional technical assumption, namely, that the background equation be noncharacteristic with respect to the shock.