Pointwise Green's function estimates toward stability for
degenerate viscous shock waves
Pointwise Green's function estimates toward stability
for degenerate viscous shock waves
We consider degenerate viscous shock waves arising in systems of two conservation
laws, where degeneracy here describes viscous shock waves for which the asymptotic
endstates are sonic to the hyperbolic system (the shock speed is equal to one of
the characteristic speeds). In particular, we develop detailed pointwise estimates
on the Green's function associated with the linearized perturbation equation,
sufficient for establishing that spectral stability implies nonlinear stability.
The analysis of degenerate viscous shock waves involves several new features,
such as algebraic (non-integrable) convection coefficients, loss of analyticity
of the Evans function at the leading eigenvalue, and time decay intermediate
between that of the Lax case and that of the undercompressive case.
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