Brinkman's law as $Γ$-limit of compressible low Mach Navier-Stokes equations and application to randomly perforated domains

Abstract: We consider the time-dependent compressible Navier-Stokes equations in the low Mach number regime inside a family of domains $(\Omega_\varepsilon){\varepsilon > 0}$ in $\mathbb{R}^3$. Assuming that $\lim{\varepsilon \to 0} \Omega_\varepsilon = \Omega \subset \mathbb{R}^3$ in a suitable sense, we show that in the limit the fluid flow inside $\Omega$ is governed by the incompressible Navier-Stokes-Brinkman equations, provided the latter one admits a strong solution. The abstract convergence result is complemented with a stochastic homogenization result for randomly perforated domains in the critical regime.