Pointwise Green's function bounds and stability of relaxation shocks.

We establish sharp pointwise Green's function bounds and consequent linearized stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([32, 108, 110]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, hyperbolic stability of the corresponding ideal shock of the associated equilibrium system, and transversality of the connecting profile, with no additional assumptions on the structure or strength of the shock. Restricting to Lax type shocks, we establish the further result of nonlinear stability with respect to small L(1) boolean AND H(2) perturbations, with sharp rates of decay in L(p), 2 less than or equal to p less than or equal to infinity, for weak shocks of general simultaneously symmetrizable systems; for discrete kinetic models, and initial perturbation small in W(3,1) boolean AND W(3,infinity), we obtain sharp rates of decay in L(p), 1 less than or equal to p less than or equal to infinity, for (Lax type) shocks of arbitrary strength. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin-Xin and Broadwell models, for which spectral stability has been established in [61, 43], and in [52], respectively. Our analysis follows the basic pointwise semigroup approach introduced by Zumbrun and Howard [107] for the study of traveling waves of parabolic systems; however, significant extensions are required to deal with the nonsectorial generator and more singular short-time behavior of the associated (hyperbolic) linearized equations. Our main technical innovation is a systematic method for refining large-frequency (short-time) estimates on the resolvent kernel, suitable in the absence of parabolic smoothing. This seems particularly interesting from the viewpoint of general linear theory, replacing the zero-order estimates of existing theory with a series expansion to arbitrary order. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.