This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Greens functions $(G_varepsilon,Pi_varepsilon)$ (and fundamental solutions $(Gamma_varepsilon, Q_varepsilon)$) 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 $Pi_varepsilon$. Also, for the first time we prove the textit{adjustable} uniform estimates (i.e., Lipschitz estimate for velocity and oscillation estimate for pressure) by making full use of the Greens functions. Via these estimates, we establish the asymptotic expansions of $G_varepsilon, abla G_varepsilon, Pi_varepsilon$ 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.
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