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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Xu, Qiang¹ 

[1] Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China

关键词: Homogenization;    Stokes systems;    Convergence rates;    Lipschitz domains;    W-1,W-p estimates;   

DOI  :  10.1016/j.jde.2017.02.040

来源: Elsevier

【 摘 要 】

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 24 an almost-sharp O(epsilon ln(r(0)/epsilon)) convergence rate in L-2 space, and a sharp O(epsilon) error estimate in L2d/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,W-P estimates uniformly down to microscopic scale e without any smoothness assumption on the coefficients, where vertical bar 1/p - 1/2 vertical bar < 1/2d + epsilon and epsilon is a positive constant independent of epsilon. Combining the local estimates, based upon VMO coefficients, consequently leads to the uniform W-1,W-P estimates. Here the proofs do not rely on the well known compactness methods. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free