Convergence rates and W¹,ᵖ estimates in homogenization theory of Stokes systems in Lipschitz domains

Original · EN

Concerned with the Stokes systems with rapidly oscillating periodic coefficients, we mainly extend the recent works in SGZWS,G to those in term of Lipschitz domains. The arguments employed here are quite different from theirs, and the basic idea comes from QX2, originally motivated by SZW2,SZW12,TS. We obtain an almost-sharp O(ε(r₀/ε)) convergence rate in L² space, and a sharp O(ε) error estimate in L²ᵈ/ᵈ⁻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 |1/p-1/2|<1/2d+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.