The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system

We study the inverse of the divergence operator on a domain $\Omega \subset R^3$ perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space $L^p(\mathrm{\Omega })$ for any $1<p<3$, with a norm independent of perforation, provided the holes are suitably small and their mutual distance suitably large. Applications are given to problems arising in homogenization of steady compressible fluid flows.