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This paper is devoted to establishing the uniform estimates and asymptoticbehaviors of the Green's functions $(G_\varepsilon,\Pi_\varepsilon)$ (andfundamental solutions $(\Gamma_\varepsilon, Q_\varepsilon)$) for the Stokessystem with periodically oscillating coefficients (including a system of linearincompressible elasticity). Particular emphasis will be placed on the newoscillation estimates for the pressure component $\Pi_\varepsilon$. Also, forthe first time we prove the \textit{adjustable} uniform estimates (i.e.,Lipschitz estimate for velocity and oscillation estimate for pressure) bymaking full use of the Green's functions. Via these estimates, we establish theasymptotic expansions of $G_\varepsilon,\nabla G_\varepsilon, \Pi_\varepsilon$and more, with a tiny loss on the errors. Some estimates obtained in this paperare new even for Stokes system with constant coefficients, and possesspotential applications in homogenization of Stokes or elasticity system.

Periodic homogenization of Green's functions for Stokes systems