Stability of Large-Amplitude Shock Profiles of General Relaxation Systems

Abstract. Building on previous analyses carried out in [24, 27], we establish L1 ∩ H2 → L nonlinear orbital stability, 1 ≤ p ≤ ∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a general class of relaxation systems that includes most models in common use, 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. In particular, our results apply to standard moment closure systems, answering a question left open in [24]. The argument combines the basic nonlinear stability argument introduced [24] with an improved “Goodman-style” weighted energy estimate similar to but substantially more delicate than that used in [27] to treat large-amplitude profiles of systems with real viscosity.