  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 $L^p(\Omega)$ 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.
