Small, medium and large shock waves for non-equilibrium radiation hydrodynamic

We examine the existence of shock profiles for a hyperbolic-elliptic system arising in radiation hydrodynamics. The algebraic-differential system for the wave profile is reduced to a standard two-dimensional form that is analyzed in details showing the existence of heteroclinic connection between the two singular points of the system for any distance between the corresponding asymptotic states of the original model. Depending on the location of these asymptotic states, the profile can be either continuous or possesses at most one point of discontinuity. Moreover, a sharp threshold relative to presence of an internal absolute maximum in the temperature profile --also called {\sf Zel'dovich spike}-- is rigourously derived.Comment: 22 pages, 3 figure

Velocity-jump processes with a finite number of speeds and their asymptotically parabolic nature

The paper examines a class of first order linear hyperbolic systems, proposed as a generalization of the Goldstein-Kac model for velocity-jump processes and determined by a finite number of speeds and corresponding transition rates. It is shown that the large-time behavior is described by a corresponding scalar diffusive equation of parabolic type, defined by a diffusion matrix for which an explicit formula is given. Such representation takes advantage of a variant of the Kirchoff's matrix tree Theorem applied to the graph associated to the system and given by considering the velocities as verteces and the transition rates as weights of the arcs

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear 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. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. 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.Comment: 120 pages. Changes since original submission. Corrected typos, esp. energy estimates of Section 7, corrected bad forward references, expanded Remark 1.17, end of introductio