Compactness and Large-Scale Regularity for Darcy's Law

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 $1<p<\infty$.