Optimal time-decay estimates for the compressible navier-stokes equations in the critical l p framework

TL;DR

This paper establishes optimal time-decay rates for solutions to the compressible Navier-Stokes equations in critical Lp spaces across multiple dimensions, extending previous results and using refined Fourier space inequalities.

Contribution

It proves the decay rates in the more general Lp critical framework, broadening the understanding of long-term behavior of solutions in critical regularity spaces.

Findings

Decay rate of t^(-d(1/p - 1/4)) for solutions as t→∞

Applicable to a range of dimensions d ≥ 2

Method based on refined Fourier space inequalities

Abstract

The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed in [7] more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in general critical spaces and any dimension d $\ge$ 2 has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of [7] but also in the more general L p critical framework of [3, 6, 14]. More precisely, we show that under a mild additional decay assumption that is satisfied if the low frequencies of the initial data are in e.g. L p/2 (R d), the L p norm (the slightly stronger $\dot B^0_{p,1}$ norm in fact) of the critical global solutions decays like t --d(1 p -- 1 4) for t $\rightarrow$ +$\infty$, exactly as firstly observed by A. Matsumura and T. Nishida in [23] in the case p = 2 and d = 3, for solutions with high Sobolev regularity. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics.