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

Abstract

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.

22 pages, final version, to appear in ESAIM: Control, Optimisation and Calculus of Variations

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