Pseudodifferential damping estimates and stability of relaxation shocks
Kevin Zumbrun
TL;DR
This work introduces a novel pseudodifferential damping framework that links nonlinear damping to high-frequency linear stability for relaxation shocks. By developing frequency-dependent damping estimates and exploiting Kreiss-type symmetrizers, the authors unify nonlinear control with linear resolvent bounds, enabling nonlinear stability analysis across smooth and discontinuous profiles. The approach encompasses small-amplitude reductions to Kawashima-type dissipativity, 1-D large-amplitude results, and multi-D treatments that address Airy-type turning points and glancing phenomena, ultimately providing a pathway to prove nonlinear stability for large-amplitude multi-D relaxation shocks. The framework also clarifies the relationship between exponential dichotomies and symmetrizers, identifies turning-point challenges, and outlines key open problems for extending nonlinear damping to more complex, discontinuous media.
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.