Low Mach number limit of the compressible Navier-Stokes system for large initial date with critical regularity on the torus

TL;DR

This paper proves that solutions of the compressible Navier-Stokes equations on a torus converge to incompressible solutions as the Mach number approaches zero, even for large initial data with critical regularity.

Contribution

It establishes convergence for large initial data with critical regularity, extending previous results to more general initial conditions.

Findings

Solutions exist as long as incompressible solutions do

Convergence to incompressible solutions as Mach number tends to zero

Applicable to large initial data with critical regularity

Abstract

We study the low Mach number limit of the compressible Navier-Stokes equations on the torus. For large initial data with critical regularity, we prove that solutions to the compressible Navier-Stokes system exist as long as the corresponding solutions to the incompressible Navier-Stokes system exist, provided that the Mach number is sufficiently small. Furthermore, we establish the convergence of solutions of the compressible system to those of the incompressible system as the Mach number tends to zero. Our approach combines high-medium-low frequency analysis of density and velocity with the solution filtering technique via acoustic wave groups. This work provides an affirmative answer to the problem posed by Danchin [Amer.J.Math.,124(2002),1153-1219]:"Does convergence hold for large data with critical regularity?"