The Green function for the Stokes system with measurable coefficients

TL;DR

This paper constructs and analyzes the Green function for the stationary Stokes system with measurable coefficients in Lipschitz domains, establishing conditions under which it exists and satisfies pointwise estimates.

Contribution

It introduces conditions for the existence and estimates of the Green function for Stokes systems with measurable coefficients, including VMO coefficients in Lipschitz and C^1 domains.

Findings

Constructed the Green function under interior Hölder continuity assumption.

Proved the Green function satisfies global pointwise estimates.

Verified conditions for Green function estimates in VMO coefficient cases.

Abstract

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^n$, $n\ge 3$. We construct the Green function in $\Omega$ under the condition $(\bf{A1})$ that weak solutions of the system enjoy interior H\"older continuity. We also prove that $(\bf{A1})$ holds, for example, when the coefficients are $\mathrm{VMO}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(\bf{A2})$ 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 $\mathrm{BMO}$ coefficients on a Reifenberg domain, we verify that $(\bf{A2})$ is satisfied when the coefficients are $\mathrm{VMO}$ and $\Omega$ is a bounded $C^1$ domain.