Homogenization of the Stokes system in a non-periodically perforated domain

TL;DR

This paper extends homogenization results for the Stokes system from periodic to non-periodic perforated domains, demonstrating convergence to Darcy's law and establishing uniform regularity and convergence estimates.

Contribution

It generalizes homogenization results for the Stokes system to non-periodic perforated domains, including existence of correctors and $H^2$-convergence estimates.

Findings

Velocity converges to Darcy's law in non-periodic domains.

Established existence of correctors and uniform regularity estimates.

Proved $H^2$-convergence estimates for specific force fields.

Abstract

In our recent work [8], we have studied the homogenization of the Poisson equation in a class of non periodically perforated domains. In this paper, we examine the case of the Stokes system. We consider a porous medium in which the characteristic distance between two holes, denoted by $\epsilon$, is proportional to the characteristic size of the holes. It is well known (see [1],[17] and [19]) that, when the holes are periodically distributed in space, the velocity converges to a limit given by the Darcy's law when the size of the holes tends to zero. We generalize these results to the setting of [8]. The non-periodic domains are defined as a local perturbation of a periodic distribution of holes. We obtain classical results of the homogenization theory in perforated domains (existence of correctors and regularity estimates uniform in $\epsilon$) and we prove $H^2$--convergence estimates for particular force fields.