Periodic homogenization of Green's functions for Stokes systems

Shu Gu, Jinping Zhuge

Published 2017-10-15Version 1

This paper is devoted to establishing the uniform regularity and asymptotic behavior of the Green's functions $(G_\varepsilon,\Pi_\varepsilon)$ (and fundamental solutions $(\Gamma_\varepsilon, Q_\varepsilon)$) for the Stokes systems with periodically oscillating coefficients. Particular emphasis will be placed upon the new oscillation estimates for the pressure component $\Pi_\varepsilon$. Also, for the first time we prove the \textit{adjustable} uniform estimates (i.e., the adjustable uniform Lipschitz estimates for the velocity and oscillation estimates for the 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. Besides their own interests, the results in this paper concerning the Green's functions lay the foundation for future applications in periodic homogenization of Stokes systems.