Inverse of divergence and homogenization of compressible Navier-Stokes equations in randomly perforated domains

Abstract

We analyze the behavior of weak solutions to compressible viscous fluid flows in a bounded domain in R3, randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like εα, α > 3, with ε denoting the average distance between the balls, the problem homogenize to the same limiting equation. Our main contribution is a construction of the Bogovski˘ı operator, uniformly in ε, without any assumptions on the minimal distance between the balls.