  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.
