Convergence rates and $W^{1,p}$ estimates in homogenization theory of Stokes systems in Lipschitz domains

TL;DR

This paper extends homogenization results for Stokes systems with oscillating coefficients to Lipschitz domains, providing sharp convergence rates and uniform $W^{1,p}$ estimates without smoothness assumptions on coefficients.

Contribution

It introduces new convergence rate estimates and $W^{1,p}$ bounds for Stokes systems in Lipschitz domains, avoiding traditional compactness methods.

Findings

Achieved almost-sharp $O(rac{ ext{epsilon}}{ ext{ln}(r_0/ ext{epsilon})})$ convergence rate in $L^2$.

Established sharp $O( ext{epsilon})$ error estimate in $L^{rac{2d}{d-1}}$ space.

Derived uniform $W^{1,p}$ estimates down to microscopic scale $ ext{epsilon}$.

Abstract

Concerned with the Stokes systems with rapidly oscillating periodic coefficients, we mainly extend the recent works in \cite{SGZWS,G} to those in term of Lipschitz domains. The arguments employed here are quite different from theirs, and the basic idea comes from \cite{QX2}, originally motivated by \cite{SZW2,SZW12,TS}. We obtain an almost-sharp $O(\varepsilon\ln(r_0/\varepsilon))$ convergence rate in $L^2$ space, and a sharp $O(\varepsilon)$ error estimate in $L^{\frac{2d}{d-1}}$ space by a little stronger assumption. Under the dimensional condition $d=2$, we also establish the optimal $O(\varepsilon)$ convergence rate on pressure terms in $L^2$ space. Then utilizing the convergence rates we can derive the $W^{1,p}$ estimates uniformly down to microscopic scale $\varepsilon$ without any smoothness assumption on the coefficients, where $|\frac{1}{p}-\frac{1}{2}|<\frac{1}{2d}+\epsilon$ and $\epsilon$ is a positive constant independent of $\varepsilon$. Combining the local estimates, based upon $\text{VMO}$ coefficients, consequently leads to the uniform $W^{1,p}$ estimates. Here the proofs do not rely on the well known compactness methods.