This work focuses on the stability theory of nonlinear traveling
waves, with an emphasis on front propagation arising in the continuum
and kinetic theories of compressible flow. We report on a recent
collection results that use both analytical and numerical techniques
as part of a general strategy for proving the stability of high Mach
number viscous shock layers. We also discuss our efforts to
numerically demonstrate stability in the intermediate Mach regime,
noting that low-Mach number stability has already been proven in broad
generality. Our technical approach centers around Evans function
computation, energy estimates, and asymptotic ODE techniques,
including the gap, tracking, and conjugation lemmata, blow-up methods,
and geometric singular perturbation theory; also included are
techniques in bifurcation theory and the spectral theory of linear
operators.
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