Optimal decay estimates in the critical $L^p$ framework for flows of compressible viscous and heat-conductive gases

TL;DR

This paper establishes optimal decay estimates for global solutions of the compressible Navier-Stokes equations in the critical $L^p$ framework, extending previous results and improving decay rates for high frequencies.

Contribution

It extends decay estimates to the full Navier-Stokes system in the critical $L^p$ setting and improves decay rates for high frequencies compared to prior work.

Findings

Recovered classical decay rate $t^{-3/4}$ in 3D.

Established decay estimates under mild integrability assumptions.

Improved decay rates for high frequencies of solutions.

Abstract

The global existence issue in critical regularity spaces for the full Navier-Stokes equationssatisfied by compressible viscous and heat-conductive gases has been first addressed in \cite{D2}, then recently extended to the general $L^p$ framework in \cite{DH}.In the present work, we establish decay estimates for the global solutions constructed in \cite{DH}, under anadditional mild integrability assumption that is satisfied if the low frequencies of the initial data are in $L^{p/2}({\mathbb{R}}^d).$As a by-product we recover in dimension three the classical decay rate $t^{-\frac34}$ for $t\to+\infty$ that has been observed by A. Matsumura and T. Nishida in \cite{MN2} for solutions with high Sobolev regularity. Compared to a recent paper of us \cite{DX} dedicated to the barotropic case, not only we are able to treat the full system, but we also improve the decay rates for the high frequencies of the solution. We believe the correspondant decay exponents to be optimal.