Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit

Abstract

We study the behaviour of the solution $u_{\epsilon }$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space ${\mathbb{R}}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size ${\epsilon }^{\alpha }=N^{-\alpha /3}$ and the viscosity ${\epsilon }^{\gamma }=N^{-\gamma /3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($\alpha +\gamma >3$) and critical ($\alpha +\gamma =3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $\alpha >2$ so that the particles do not overlap with overwhelming probability.