The Green function for the Stokes system with measurable coefficients
Jongkeun Choi Ki-Ahm Lee
Communications on Pure & Applied Analysis 2017, 16(6): 1989-2022 doi: 10.3934/cpaa.2017098

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $Ω\subset \mathbb{R}^n$, $n≥ 3$. We construct the Green function in $Ω$ under the condition $(\bf{A1})$ that weak solutions of the system enjoy interior Hölder 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 $Ω$ is a bounded $C^1$ domain.

keywords: Stokes system Green function $L^q$-estimate $\mathrm{VMO}$ coefficients Reifenberg flat domain

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