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

TL;DR

This paper constructs an inverse divergence operator on perforated domains, demonstrating its boundedness in certain Lebesgue spaces, with applications to homogenization in compressible fluid flow models.

Contribution

It introduces a method to invert divergence on perforated domains with bounds independent of perforation size, advancing homogenization techniques for fluid dynamics.

Findings

Inverse divergence operator exists on perforated domains for 1<p<3.

Operator norm is independent of perforation details under certain conditions.

Applications to homogenization of steady compressible Navier-Stokes flows.

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.