Optimal Boundary Estimates for Stokes Systems in Homogenization Theory
Shu Gu, Qiang Xu · SIAM Journal on Mathematical Analysis · 2017
This paper is concerned with sharp boundary regularity estimates in homogenization of the Dirichlet problem for Stokes systems. We obtain Lipschitz estimates for the velocity and $L^\infty$ estimate for the pressure, under some reasonable smoothness assumption on rapidly oscillating periodic coefficients. We find a new way to obtain the sharp uniform boundary estimates without imposing the symmetry assumption on coefficients. Additionally, we emphasize that the $L^\infty$ estimate for the pressure does require the $O(\varepsilon^{1/2})$ convergence rate, locally at least, compared to $O(\varepsilon^\lambda)$ for the velocity with $\lambda\in(0,1/2)$.