The Green function for the Stokes system with measurable coefficients

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $\Omega\subset $\mathbb{R}$^n$,$n\ge 3$\mathrm{.}WeconstructtheGreenfunctionin$\Omega$ under the condition $(\mathbf{A}\mathbf{1})$ that weak solutions of the system enjoy interior H"older continuity. We also prove that $(\mathbf{A}\mathbf{1})$ holds, for example, when the coefficients are $\mathrm{V}\mathrm{M}\mathrm{O}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(\mathbf{A}\mathbf{2})$ that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori $L^q$-estimates for Stokes systems with $\mathrm{B}\mathrm{M}\mathrm{O}$ coefficients on a Reifenberg domain, we verify that $(\mathbf{A}\mathbf{2})$ is satisfied when the coefficients are $\mathrm{V}\mathrm{M}\mathrm{O}$ and $\mathrm{\Omega }$ is a bounded $C^1$ domain.