Pointwise semigroup methods and stability of viscous shock waves
Pointwise semigroup methods and stability of viscous shock waves
Considered as rest points of ODE on L^p, stationary viscous shock
waves present a critical case for which standard semigroup methods to
not suffice to determine stability. More precisely, there is no
spectral gap between stationary modes and essential spectrum of
the linearized operator about the wave, a fact that precludes the
usual analysis by decomposition into invariant subspaces. For
this reason, there have been until recently no results on shock
stability from the semigroup perspective except in the scalar
or totally compressive case, each of which can be reduced to the
standard semigroup setting by Sattinger's method of weighted norms.
We overcome this difficulty in the general case by the introduction
of new, pointwise semigroup techniques, generalizing earlier
work of Howard, Kapitula, and Zeng. These techniques allow us to
do "hard" analysis in PDE within the dynamical systems/semigroup
framework: in particular, to obtain sharp, global pointwise bounds
on the Green's function of the linearized operator around the
wave, sufficient of the analysis of linear and nonlinear stability.
The method is general and should find applications also in other
situations of sensitive stability.
Central to our analysis is a notion of "effective" point spectrum
that can be extended to regions of essential spectrum. This turns out
to be intimately related to the Evans function, a well-known tool
for the spectral analysis of traveling waves. Indeed, crucial to our
whole analysis is the "Gap Lemma", a technical result developed
originally in the context of Evans function theory. Using these
new tools, we can treat over- and undercompressive, and even
strong shock waves for systems within the same framework used for
standard weak (i.e., slowly varying) Lax waves. In all cases, we
show that stability is determined by the simple and numerically
computable condition that the number of zeros of the Evans function
in the right complex half-plane be equal to the dimension of the
stationary manifold of nearby traveling wave solutions. Interpreting
this criterion in the conservation law setting, we quickly recover
all known analytic stability results, obtaining several new results
as well.
Return to Publications page