Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes

TL;DR

This paper proves that in a perforated domain with many tiny holes, the solutions of the compressible Navier-Stokes equations converge to those in a smooth domain, extending previous results to the unsteady, barotropic case.

Contribution

It is the first to establish homogenization results for the unsteady, barotropic compressible Navier-Stokes equations in perforated domains with tiny holes.

Findings

Homogenized equations match the original in the domain without holes.

Results extend previous work from Stokes and stationary Navier-Stokes to unsteady case.

Main technical advance is the analysis of the Bogovskii operator in non-Lipschitz domains.

Abstract

We consider the homogenization problem of the compressible Navier-Stokes equations in a bounded three dimensional domain perforated with very tiny holes. As the number of holes increases to infinity, we show that, if the size of the holes is small enough, the homogenized equations are the same as the compressible Navier-Stokes equations in the homogeneous domain---domain without holes. This coincides with the previous studies for the Stokes equations and the stationary Navier-Stokes equations. It is the first result of this kind in the instationary barotropic compressible setting. The main technical novelty is the study of the Bogovskii operator in non-Lipschitz domains.