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 < ∞.

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.

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This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green's functions (Gε, Πε) (and fundamental solutions (Γε, Qε)) for the Stokes system with periodically oscillating coefficients (including a system of linear incompressible elasticity). Particular emphasis will be placed on the new oscillation estimates for the pressure component Πε. Also, for the first time we prove the adjustable uniform estimates (i.e., Lipschitz estimate for velocity and oscillation estimate for pressure) by making full use of the Green's functions. Via these estimates, we establish the asymptotic expansions of Gε, ∇Gε, Πε and more, with a tiny loss on the errors. Some estimates obtained in this paper are new even for Stokes system with constant coefficients, and possess potential applications in homogenization of Stokes or elasticity system.2010 Mathematics Subject Classification. 35B27, 76M50, 35C20.

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