Stability with respect to domain of the low Mach number limit of compressible viscous fluids

TL;DR

This paper investigates how solutions to the compressible Navier-Stokes equations behave as the Mach number approaches zero, especially in varying exterior domains, showing convergence to incompressible flow with strong local velocity limits.

Contribution

It provides a rigorous analysis of the low Mach number limit in variable exterior domains, demonstrating stability and convergence of solutions to incompressible flow, using spectral analysis techniques.

Findings

Velocity converges strongly to incompressible flow on compact sets.

Density becomes constant in the zero Mach limit.

Convergence is uniform across a class of domains.

Abstract

We study the asymptotic limit of solutions to the barotropic Navier-Stokes system, when the Mach number is proportional to a small parameter $\ep \to 0$ and the fluid is confined to an exterior spatial domain $\Omega_\ep$ that may vary with $\ep$. As $\epsilon \rightarrow 0$, it is shown that the fluid density becomes constant while the velocity converges to a solenoidal vector field satisfying the incompressible Navier-Stokes equations on a limit domain. The velocities approach the limit strongly (a.a.) on any compact set, uniformly with respect to a certain class of domains. The proof is based on spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.