Pseudodifferential damping estimates and stability of relaxation shocks

TL;DR

This paper introduces a novel approach using frequency-dependent pseudodifferential damping estimates to analyze the stability of relaxation shocks, offering a simplified and sharp alternative to traditional energy methods.

Contribution

It proposes an innovative pseudodifferential framework for damping estimates, extending stability analysis techniques for relaxation shocks beyond classical methods.

Findings

Equivalent to high-frequency spectral stability in linear case

Provides sharp, pointwise damping estimates

Simplifies the analysis of shock stability

Abstract

A bottleneck in the theory of large-amplitude and multi-d viscous and relaxation shock stability is the development of nonlinear damping estimates controlling higher by lower derivatives. These have traditionally proceeded from time-evolution bounds based on Friedrichs symmetric and Kawashima or Goodman type energy estimates. Here, we propose an alternative program based on frequency-dependent pseudodifferential time-space damping estimates in the spirit of Kreiss. These are seen to be equivalent in the linear case to high-frequency spectral stability, and, just as for the constant-coefficient analysis of Kreiss, sharp in a pointwise, fixed-frequency, sense. This point of view leads to a number of simplifications and extensions using already-existing analysis. We point to the new issue of turning points, analogous to glancing points in the constant-coefficient case as an important direction for further development.