Homogenization of Stokes Systems and Uniform Regularity Estimates

TL;DR

This paper develops uniform regularity estimates for Stokes systems with rapidly oscillating periodic coefficients, providing interior Lipschitz bounds, pressure estimates, and boundary regularity results.

Contribution

It establishes new uniform regularity estimates for Stokes systems with oscillating coefficients, including interior and boundary bounds, and a Liouville property.

Findings

Interior Lipschitz estimates for velocity

L^ estimates for pressure

Boundary W^{1,p} estimates in C^1 domains

Abstract

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and $L^\infty$ estimates for the pressure as well as a Liouville property for solutions in $\mathbb{R}^d$. We also obtain the boundary $W^{1,p}$ estimates in a bounded $C^1$ domain for any $1<p<\infty$.