  This paper is devoted to establishing the uniform estimates and asymptotic
behaviors of the Green's 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 Green's functions. Via these estimates, we establish the
asymptotic expansions of $G_\varepsilon,\nabla G_\varepsilon, \Pi_\varepsilon$
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.
