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Green Functions of Conormal Derivative Problems for Stationary Stokes System

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Abstract

We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.

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Acknowledgements

The authors would like to thank the referees for careful reading and helpful comments.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Jongkeun Choi & Hongjie Dong

  2. Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea

    Doyoon Kim

Authors
  1. Jongkeun Choi
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  2. Hongjie Dong
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  3. Doyoon Kim
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Correspondence to Doyoon Kim.

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The authors declare that they have no competing interests.

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Communicated by R. Shvydkoy

Jongkeun Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A6A3A03005168). Hongjie Dong was partially supported by the NSF under agreement DMS-1600593. Doyoon Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03934369).

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Choi, J., Dong, H. & Kim, D. Green Functions of Conormal Derivative Problems for Stationary Stokes System. J. Math. Fluid Mech. 20, 1745–1769 (2018). https://doi.org/10.1007/s00021-018-0387-0

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