Uniform regularity for the compressible Navier-Stokes system with low Mach number in bounded domains

TL;DR

This paper proves uniform regularity estimates for the compressible Navier-Stokes equations with low Mach number in bounded domains, enabling the justification of the incompressible limit even with ill-prepared initial data.

Contribution

It introduces an anisotropic functional framework to handle boundary layer effects and establishes uniform estimates for the system with ill-prepared initial data.

Findings

Uniform estimates independent of Mach number

Local existence of strong solutions

Justification of incompressible limit

Abstract

We establish uniform with respect to the Mach number regularity estimates for the isentropic compressible Navier-Stokes system in smooth domains with Navier-slip condition on the boundary in the general case of ill-prepared initial data. To match the boundary layer effects due to the fast oscillations and the ill-prepared initial data assumption, we prove uniform estimates in an anisotropic functional framework with only one normal derivative close to the boundary. This allows to prove the local existence of a strong solution on a time interval independent of the Mach number and to justify the incompressible limit through a simple compactness argument.