The Green function for the Stokes system with measurable coefficients

Jongkeun Choi; Ki-Ahm Lee

doi:10.3934/cpaa.2017098

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2017, Volume 16, Issue 6: 1989-2022. Doi: 10.3934/cpaa.2017098

This issue Previous Article Almost reducibility of linear difference systems from a spectral point of view Next Article Two-and multi-phase quadrature surfaces

The Green function for the Stokes system with measurable coefficients

Jongkeun Choi^1, and
Ki-Ahm Lee^2,3, ,

1.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
2.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
3.
Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea

* Corresponding author
* Corresponding author

Received: September 2016

Revised: April 2017

Published: September 2017

J. Choi was supported by BK21 PLUS SNU Mathematical Sciences Division. Ki-Ahm Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2015R1A4A1041675).

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J08; Secondary: 35J57, 35R05.

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