Stability of Undercompressive Shock Profiles
Stability of Undercompressive Shock Profiles
Using a simplified pointwise iteration scheme,
we establish nonlinear phase-asymptotic orbital stability of
large-amplitude Lax, undercompressive, overcompressive,
and mixed under--overcompressive type shock profiles
of strictly parabolic systems of conservation laws
with respect to initial perturbations $|u_0(x)|\le E_0 (1+|x|)^{-3/2}$
in $C^{0+\alpha}$, $E_0$ sufficiently small,
under the necessary conditions of spectral and hyperbolic
stability together with transversality of the connecting profile.
This completes the program initiated by Zumbrun and Howard in \cite{ZH},
extending to the general undercompressive case results obtained for Lax
and overcompressive shock profiles in \cite{SzX}, \cite{L},
\cite{ZH}, \cite{Z.2}, \cite{Ra}, \cite{MZ.1}--\cite{MZ.5},
and for special undercompressive profiles in
\cite{LZ.1}--\cite{LZ.2}, \cite{HZ}.
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