Global existence and optimal decay rates of strong solutions to the equilibrium diffusion model arising in radiation hydrodynamics

Abstract

In this paper, we investigate the global existence and optimal decay rates of strong solutions in the critical Sobolev space $H^2\left(R^3\right)$ to the equilibrium diffusion model arising in radiation hydrodynamics. This model is composed of the full compressible Navier-Stokes equations with radiation diffusion terms. Assuming that the initial data $(n_0,u_0,{\theta }_0)$ is sufficiently small in $H^2$-norm, we establish the global existence of strong solutions near the equilibrium state $(1,0,1)$ by applying the refined energy method. Then, by performing Fourier analysis techniques and exploiting the frequency decomposition method, we get the optimal time-decay rates of strong solutions (including the highest order spatial derivatives) in $L^2$-norm. In particular, the lower bound of time-decay rates of strong solutions is also obtained by making use of Hodge decomposition, delicate spectral analysis and the theory of Besov space. In addition, we obtain the exponential decay of strong solutions in the periodic domain case.