Compactness and Large-Scale Regularity for Darcy's Law

TL;DR

This paper develops quantitative homogenization techniques for the steady Stokes equations in perforated domains, establishing large-scale regularity and uniform estimates for velocity and pressure, advancing understanding of fluid flow in complex media.

Contribution

It introduces a compactness method to derive large-scale regularity estimates for the Stokes equations in periodically perforated domains, providing new uniform Lipschitz and Sobolev estimates.

Findings

Established large-scale $C^{1, \alpha}$ and Lipschitz estimates for velocity.

Derived uniform $W^{k, p}$ estimates for pressure.

Provided a framework for quantitative homogenization in perforated domains.

Abstract

This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior $C^{1, \alpha}$ and Lipschitz estimates for the velocity as well as the corresponding estimates for the pressure. These estimates, when combined with the classical regularity estimates for the Stokes equations, yield the uniform Lipschitz estimates. As a consequence, we also obtain the uniform $W^{k, p}$ estimates for $1<p<\infty$.