Dissipative structure and decay rate for an inviscid non-equilibrium radiation hydrodynamics system

TL;DR

This paper analyzes a non-equilibrium radiation hydrodynamics model, proving local and global existence of solutions, establishing decay rates, and introducing an entropy function to symmetrize and study the system's behavior.

Contribution

It demonstrates the existence of solutions in multiple dimensions, identifies the genuinely coupled nature of the model in one dimension, and introduces an entropy function for symmetrization.

Findings

Local solutions exist in several space dimensions.

The one-dimensional model is genuinely coupled.

Linear decay rates are established for the one-dimensional system.

Abstract

This paper studies the diffusion approximation, non-equilibrium model of radiation hydrodynamics derived by Buet and Despr\'es (J. Quant. Spectrosc. Radiat. Transf. 85 (2004), no. 3-4, 385-418). The latter describes a non-relativistic inviscid fluid subject to a radiative field under the non-equilibrium hypothesis, that is, when the temperature of the fluid is different from the radiation temperature. It is shown that local solutions exist for the general system in several space dimensions. It is also proved that only the one-dimensional model is genuinely coupled in the sense of Kawashima and Shizuta (Hokkaido Math. J. 14 (1985), no. 2, 249-275). A notion of entropy function for non-conservative parabolic balance laws is also introduced. It is shown that the entropy identified by Buet and Despr\'es is an entropy function for the system in the latter sense. This entropy is used to recast the one-dimensional system in terms of a new set of perturbation variables and to symmetrize it. With the aid of genuine coupling and symmetrization, linear decay rates are obtained for the one dimensional problem. These estimates, combined with the local existence result, yield the global existence and decay in time of perturbations of constant equilibrium states in one space dimension.