Dissipative structure and decay rate for an inviscid non-equilibrium radiation hydrodynamics system
Corrado Lattanzio, Ramón G. Plaza, José Manuel Valdovinos
TL;DR
This work analyzes the inviscid, non-equilibrium diffusion radiation hydrodynamics model of Buet and Després for a non-relativistic fluid under radiative influence. It develops an entropy-based symmetrization using the Buet–Després entropy and introduces a new perturbation variable to reveal the dissipative structure, establishing linear decay via genuine coupling in 1D. By combining local existence with nonlinear energy estimates, the authors prove global well-posedness and time decay for small perturbations of a constant equilibrium in one space dimension, with a decay rate of $\|V-\overline{V}\|_{s-1} \lesssim (1+t)^{-1/4}$. They also show that the system is not genuinely coupled in dimensions $d\ge 2$ (without damping/viscosity), indicating the need for additional dissipative effects for multi-dimensional stability. The entropy-based, Kawashima–Shizuta framework provides a robust approach to non-conservative viscous balance systems and offers insights for extending the analysis to more complex radiation-hydrodynamics models.
Abstract
This paper studies the diffusion approximation, non-equilibrium model of radiation hydrodynamics derived by Buet and Després (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 Després 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.