Pseudodifferential damping estimates and stability of relaxation shocks
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.
Cite
@article{arxiv.2407.14484,
title = {Pseudodifferential damping estimates and stability of relaxation shocks},
author = {Kevin Zumbrun},
journal= {arXiv preprint arXiv:2407.14484},
year = {2026}
}