Optimal time-decay for Euler-Fourier system with damping in the critical $L^2$ framework

TL;DR

This paper establishes optimal decay rates for solutions to the Euler-Fourier system with damping in any spatial dimension using a novel $L^2$ energy method that removes the need for small initial data.

Contribution

It introduces a time-weighted energy approach in the $L^2$ framework and a Lyapunov functional analysis to achieve optimal decay without smallness assumptions.

Findings

Derived optimal time-decay rates for solutions.

Removed smallness condition on initial data.

Identified a damped mode with faster decay.

Abstract

This paper is concerned with the large time behavior of solutions to the Euler-Fourier system with damping in $\mathbb{R}^{d}~(d\geq1)$. A time-weighted energy argument has been developed within the $L^2$ framework to derive the optimal time-decay rates, which enables us to remove the smallness of low-frequencies of initial data. A great part of our analysis relies on the study of a Lyapunov functional in the spirit of [13], which mainly depends on some elaborate use of non-classical Besov product estimates and interpolations. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.