Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

TL;DR

This paper establishes sharp boundary regularity estimates for Stokes systems in homogenization, providing Lipschitz bounds for velocity and $L^ abla$ bounds for pressure without symmetry assumptions on coefficients.

Contribution

It introduces a new approach to obtain uniform boundary estimates for Stokes systems in homogenization, avoiding Rellich estimates and symmetry assumptions.

Findings

Lipschitz estimates for velocity in homogenized Stokes systems

$L^ abla$ estimates for pressure without symmetry assumptions

Pressure estimate requires $O( ext{}\varepsilon^{1/2})$ convergence rate

Abstract

The paper concerns the sharp boundary regularity estimates in homogenization of Dirichlet problem for Stokes systems. We obtain the Lipschitz estimates for velocity term and $L^\infty$ estimate for pressure term, under some reasonable smoothness assumption on rapidly oscillating periodic coefficients. The approach is based on convergence rates, originally investigated by S. Armstrong and Z. Shen in \cite{SZ,SZW12}, however the argument developed here does not rely on the Rellich estimates. In this sense, we find a new way to obtain the sharp uniform boundary estimates without imposing the symmetry assumption on coefficients. Additionally, we emphasize that $L^\infty$ estimate for the pressure term does require the $O(\varepsilon^{1/2})$ convergence rate, locally at least, compared to $O(\varepsilon^\lambda)$ for the velocity term, where $\lambda\in(0,1/2)$.