Global well-posedness and optimal decay rates of classical solutions to the compressible Navier-Stokes-Fourier-P$_1$ approximation model in radiation hydrodynamics

TL;DR

This paper proves the global existence and decay rates of classical solutions to a coupled fluid-radiation model in three dimensions, using Fourier analysis and energy estimates, marking the first such result for this model.

Contribution

It introduces a new method to establish global well-posedness and optimal decay rates for the NSF-$P_1$ radiation hydrodynamics model with small initial perturbations.

Findings

Global well-posedness of classical solutions is established.

Optimal decay rates in $L^p$ norms are derived.

A novel approach overcomes difficulties from radiation-related linear terms.

Abstract

In this paper, the compressible Navier-Stokes-Fourier-$P_1$ (NSF-$P_1$) approximation model in radiation hydrodynamics is investigated in the whole space $\mathbb{R}^3$. This model consists of the compressible NSF equations of fluid coupled with the transport equations of the radiation field propagation. Assuming that the initial data are a small perturbation near the equilibrium state, we establish the global well-posedness of classical solutions for this model by performing the Fourier analysis techniques and employing the delicate energy estimates in frequency spaces. Here, we develop a new method to overcome a series of difficulties arising from the linear terms $n_1$ in (3.2)$_2$ and $n_0$ in (3.3)$_3$ related to the radiation intensity. Furthermore, if the $L^1$-norm of the initial data is bounded, we obtain the optimal time decay rates of the classical solution at $L^p$-norm $(2\leq p\leq \infty)$. To the best of our knowledge, this is the first result on the global well-posedness of the NSF-$P_1$ approximation model.