Convergence rates and W1,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 [19], [20] to those in term of Lipschitz domains. The arguments employed here are quite different from theirs, and the basic idea comes from [37], originally motivated by [23], [27], [33]. We obtain an almost-sharp O(εln⁡(r0/ε)) convergence rate in L2 space, and a sharp O(ε) error estimate in L2dd−1 space by a little stronger assumption. Under the dimensional condition d=2, we also establish the optimal O(ε) convergence rate on pressure terms in L2 space. Then utilizing the convergence rates we can derive the W1,p estimates uniformly down to microscopic scale ε without any smoothness assumption on the coefficients, where |1p−12|<12d+ϵ and ϵ is a positive constant independent of ε. Combining the local estimates, based upon VMO coefficients, consequently leads to the uniform W1,p estimates. Here the proofs do not rely on the well known compactness methods.