DECAY RATES TO VISCOUS CONTACT WAVES FOR A 1D COMPRESSIBLE RADIATION HYDRODYNAMICS MODEL

Abstract

We are concerned with the nonlinear stability of contact waves subject to general perturbations of initial datum in the Cauchy problem for a radiating gas model. The model is represented mathematically as a one-dimensional hyperbolic–elliptic system. It is known that general perturbations of contact discontinuities may generate diffusion waves which evolve and interact with the contact wave. In order to quantify the decay to the contact waves exactly, we need to construct the corresponding diffusion waves explicitly depending on the perturbation of the initial datum. Then, the constructed diffusion waves can be added to the viscous contact wave to form a new combined wave pattern. In this paper, we will show that the new combined wave pattern is nonlinear stable provided that the general perturbation of the initial datum and the strength of the contact wave are suitably small. In particular we give detailed convergence rates using anti-derivative methods and elaborated energy estimates. The work extends the results of Huang, Xin and Yang in [Contact discontinuity with general perturbations for gas motions, Adv. Math. 219 (2008) 1246–1297] for compressible Navier–Stokes equations to a system with much weaker dissipation mechanism.