Homogenization of Stokes Systems and Uniform Regularity Estimates

Abstract

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L ∞ estimates for the pressure as well as a Liouville property for solutions in R d . We also obtain the boundary W 1,p estimates in a bounded C 1 domain for any 1 < p < ∞.

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

“…Then it is well known that u ε ⇀ u 0 weakly in H 1 (Ω; R d ) and p ε − − Ω p ε ⇀ p 0 − − Ω p 0 weakly in L 2 (Ω) as ε → 0 (see for example [5,20]), where the notation − Ω denotes the average integral over Ω, and the pair (u 0 , p 0 ) ∈ H 1 (Ω;…”

Uniform boundary regularity in almost-periodic homogenization

“…The second application is devoted to the large scale Rellich estimate in L 2 . More precisely, we will show that 14) and 15) for all r ≥ ω k,σ (ε). These estimates imply the usual Rellich estimate if ω k,σ (ε) = O(ε) and A possesses symmetry and certain smoothness; see the remark after Theorem 5.3.…”

“…The first breakthrough was due to [6] in which the authors proved the uniform Lipschitz estimates for Dirichlet problems with periodic coefficients by a compactness argument originating from the regularity theory in the calculus of variations and minimal surfaces. The compactness argument has been proved extremely useful and extensively applied in all kinds of homogenization problems; see [7,11,14,16,18] for more references on this topic. However, the Lipschitz estimate for Neumann problems was not known until recent remarkable work in [18], where the compactness argument was used along with a delicate iteration scheme.…”

Periodic homogenization of Green’s functions for Stokes systems

“…In this section, we give a review of homogenization theory of the Stokes systems with periodic coefficients. We refer the reader to [7, pp.76-81] and [18,17] for more details. We begin with the solvability of Stokes system (1.1) and the energy estimate.…”

“…To handle u (1) ε , for any t > 0 and x ∈ D such that D 2t (x) ⊂ D, we apply the uniform Hölder estimate (see [18]) to obtain,…”