Incompressible limit for the free surface Navier-Stokes system

TL;DR

This paper proves that as the Mach number approaches zero, the compressible free surface Navier-Stokes system converges to the incompressible system, with uniform regularity estimates ensuring a rigorous low Mach number limit.

Contribution

It provides the first uniform regularity estimates for the free surface compressible Navier-Stokes system in three dimensions, justifying the low Mach number limit under well-prepared initial data.

Findings

Uniform estimates with respect to Mach number

Convergence to incompressible system in low Mach limit

Control of boundary layer oscillations

Abstract

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $\sqrt{\varepsilon}$, $\varepsilon$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.