### Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems

Abstract: We establish nonlinear $L^1\cap H^3\to L^p$ orbital stability, $2\le p\le \infty$, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas- and magneto-hydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [MN], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier-Stokes equations for a gamma-law gas as $\gamma\to 1$: the first complete large-amplitude stability result for a shock profile of a system with real (i.e. partial) viscosity.

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