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

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.