Convergence rates in homogenization of Stokes systems

Abstract

This paper studies the convergence rates in L 2 and H 1 of Dirichelt problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.

Periodic homogenization of Green’s functions for Stokes systems

“…Recently, notable progress has been made towards the theory of convergence rates and uniform regularity in homogenization of Stokes system (1.1); see [18,17,19,32,1,9]. In the present paper, among others, we are particularly interested in the asymptotic behavior of the Green's functions and their derivatives for the 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.…”

Uniform boundary regularity in almost-periodic homogenization

“…Based on the flatness property of weak solutions of L 0 + λ, we are able to prove the following flatness property of u ε : 13) for some fixed 0 < θ < 1 and all ε < r < 1, where H and Φ are defined in (4.2) and (4.1), respectively. Notice that H(r; u ε ) quantifies the local regularity property of u ε and the second term on the right-hand side of (1.13) is the error term between u ε and u 0 .…”

“…whereq > 2 is the exponent related to the reverse Hölder estimate (Meyers' estimate) of solutions of elliptic operators, which depends only on d, m and The importance of approximate correctors is due to the fact that 13) and thus χ T could be regarded as an approximation of the usual correctors.…”

“…Recently, the sharp rates of convergence in C 1,1 domains were obtained for variational elliptic problems with rough periodic coefficients; see [12,24,25,22,13] for example. Theorem 2.11 possibly gives the nearly sharp rate of convergence for elliptic systems with almost-periodic coefficients in H. Weyl's sense.…”