Stability of small‐amplitude shock profiles of symmetric hyperbolic‐parabolic systems

Combining pointwise Green's function bounds obtained in a companion paper 36] with earlier, spectral stability results obtained in 16], we establish nonlinear orbital stability of small‐amplitude Lax‐type viscous shock profiles for the class of dissipative symmetric hyperbolic‐parabolic systems identified by Kawashima 20], notably including compressible Navier‐Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in L^p with respect to small L¹ ∩ H³ perturbations, 2 ≤ p ≤ ∞. Our analysis extends and somewhat refines the approach introduced in 35] to treat stability of relaxation profiles. © 2004 Wiley Periodicals, Inc.