Preprint 2001-027

Pointwise Green's Function Bounds and Stability of Relaxation Shocks

Corrado Mascia and Kevin Zumbrun

Abstract: 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 ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system, 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 $L^1\cap H^2$ perturbations, with sharp rates of decay in $L^p$, $2\le p\le \infty$, for weak shocks of general simultaneously symmetrizable systems; for discrete kinetic models, and initial perturbation small in $W^{3,1}\cap W^{3,\infty}$, we obtain sharp rates of decay in $L^p$, $1\le p\le \infty$, 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 [HL,JH] and [KM], respectively. Our analysis follows the basic pointwise semigroup approach introduced by Zumbrun and Howard [ZH] 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.

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Corrado Mascia, <>
Kevin Zumbrun, <>
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<> July 17 2001.

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