Decay of a model system of radiating gas

TL;DR

This paper establishes optimal decay rates for solutions of a radiating gas model in multiple dimensions without requiring small initial perturbations, using Fourier splitting and energy methods.

Contribution

It provides the first large-time decay estimates for classical solutions to the radiating gas system without smallness assumptions on initial data.

Findings

Optimal $H^N$-norm decay rates in $ ext{R}^n$ for $1 ext{-}4$ dimensions.

Optimal $L^p$-$L^2$ decay estimates for derivatives in $ ext{R}^3$.

Decay estimates achieved via Fourier splitting and refined energy methods.

Abstract

This paper is concerned with optimal time-decay estimates of solutions of the Cauchy problem to a model system of the radiating gas in $\mathbb{R}^n$. Compared to Liu and Kawashima (2011) \cite{Liu1} and Wang and Wang (2009) \cite{Wang}, without smallness assumption of initial perturbation in $L^1$-norm, we study large time behavior of small amplitude classical solutions to the Cauchy problem. The optimal $H^N$-norm time-decay rates of the solutions in $\mathbb{R}^n$ with $1\leq n\leq4$ are obtained by applying the Fourier splitting method introduced in Schonbek (1980) \cite{Schonbek1} with a slight modification and an energy method. Furthermore, basing on a refined pure energy method introduced in Guo and Wang \cite{Guo} (2011), we give optimal $L^p$-$L^2(\mathbb{R}^3)$ decay estimates of the derivatives of solutions when initial perturbation is bounded in $L^p$-norm with some $p\in(1,2]$.