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
Conormal derivative problems for stationary Stokes system in Sobolev spaces
Jongkeun Choi Hongjie Dong Doyoon Kim
Discrete & Continuous Dynamical Systems - A 2018, 38(5): 2349-2374 doi: 10.3934/dcds.2018097

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

keywords: Stokes system Reifenberg flat domains measurable coefficients conormal derivative boundary condition

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