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

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 $O(\epsilon \ln (r_0\mathrm{/}\epsilon ))$ convergence rate in $L^2$ space, and a sharp $O(\epsilon )$ 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 $O(\epsilon )$ 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 $\epsilon$ without any smoothness assumption on the coefficients, where $\mathrm{\mid }\frac{1}{p}-\frac{1}{2}\mathrm{\mid }<\frac{1}{2d}+ϵ$ and $ϵ$ is a positive constant independent of $\epsilon$. 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.