The incompressible limit in $L^p$ type critical spaces

TL;DR

This paper rigorously justifies the convergence of viscous compressible flows to incompressible Navier-Stokes equations in $L^p$ critical spaces, extending previous $L^2$ results to a broader functional framework.

Contribution

It introduces a novel $L^p$ critical regularity framework for low Mach number convergence, including ill-prepared data and heat conducting fluids.

Findings

Convergence established in $L^p$ critical Besov spaces.

Extension of results to heat conducting fluids.

Requires $L^2$ bounds on low frequencies for potential velocity and density.

Abstract

This paper aims at justifying the low Mach number convergence to the incompressible Navier-Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to $L^2$ spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the $L^p$ type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space $\dot B^{d/p-1}_{p,r}\cap\dot B^{-1}_{\infty,1}$ for some suitable $(p,r)\in[2,4]\times[1,+\infty].$ We still require $L^2$ type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier-Stokes system for heat conducting fluids.