The low mach number limit of global solutions to the full compressible Navier-Stokes system in critical Besov spaces with large initial data

TL;DR

This paper proves the global existence and convergence to incompressible models of solutions to the full compressible Navier-Stokes equations in critical Besov spaces, even with large initial data and small Mach number.

Contribution

It extends previous results by establishing global solutions for large initial data and rigorously justifies the low Mach number limit at critical regularity levels.

Findings

Global existence of solutions with large initial data

Convergence to incompressible models as Mach number tends to zero

Analysis of frequency components of density, velocity, and temperature

Abstract

We are concerned with global existence of regular solutions to full compressible Navier-Stokes equations and their asymptotic behavior when the Mach number is sufficiently small. We establish global existence in critical Besov spaces for arbitrary large initial date provided that the divergence-free component of initial velocity and the difference between initial temperature and density generate a global regular solution to incompressible Boussinesq systems. Moreover, we rigorously justify the convergence to the incompressible model as the Mach number tends to zero. The proof relies on a fine-grained analysis of the high-middle-low frequencies of density, velocity and temperature. Our result can be seen as an improvement on Danchin and He [Math. Ann., 366 (2016), no. 3-4, pp. 1365-1402], including the extension from small initial data to large initial data and new convergence results which hold at the level of critical regularity.