$\Gamma$-convergence for nearly incompressible fluids

TL;DR

This paper proves that as the Mach number approaches zero, solutions of the compressible Navier--Stokes equations in varying domains converge to solutions of the incompressible Navier--Stokes equations, using $ ext{Gamma}$-convergence techniques.

Contribution

It establishes the $ ext{Gamma}$-convergence of the compressible to incompressible Navier--Stokes equations in domains converging in the Mosco sense.

Findings

Convergence of solutions in varying domains

Validation of incompressible limit in low Mach regime

Application of $ ext{Gamma}$-convergence to fluid dynamics

Abstract

We consider the time-dependent compressible Navier--Stokes equations in the low Mach number regime in a family of domains $\Omega_\epsilon \subset R^d$ converging in the sense of Mosco to a domain $\Omega \subset R^d$, $d \in \{2,3\}$. We show the limit is the incompressible Navier--Stokes system in $\Omega$.