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

¹ Institute of Mathematics, Universität Osnabrück, Albrechtstr. 28a, 49076 Osnabrück, Germany
lars.diening@uni-osnabrueck.de
² Institute of Mathematics of the Academy of Sciences of the Czech Republic, Zitná 25, 115 67 Praha 1, Czech Republic
feireisl@math.cas.cz
³ Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic
luyong@karlin.mff.cuni.cz

Received: 29 September 2015
Revised: 12 January 2016

Abstract

We study the inverse of the divergence operator on a domain Ω ⊂ R³ perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space L^p(Ω) 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.