East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE OF GREEN FUNCTIONS FOR STOKES SYSTEMS WITH NEUMANN BOUNDARY CONDITIONS

  • Jongkeun Choi (Department of Mathematics Education, Pusan National University)
  • Received : 2023.01.18
  • Accepted : 2023.03.08
  • Published : 2023.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

We establish the existence and uniqueness of Green functions in Lipschitz domains for stationary Stokes systems with Neumann boundary conditions. For the uniqueness, we impose a different normalization condition from that in Choi et al. (J. Math. Fluid Mech., 20(4):1745-1769, 2018).

Keywords

Acknowledgement

This work was supported by a 2-Year Research Grant of Pusan National University.

References

  1. Gabriel Acosta, Ricardo G. Duran, and Maria A. Muschietti. Solutions of the divergence operator on John domains. Adv. Math., 206(2):373-401, 2006. https://doi.org/10.1016/j.aim.2005.09.004
  2. Michal Benes and Petr Kucera. Solutions to the Navier-Stokes equations with mixed boundary conditions in two-dimensional bounded domains. Math. Nachr., 289(2-3):194-212, 2016. https://doi.org/10.1002/mana.201400046
  3. Jongkeun Choi, Hongjie Dong, and Doyoon Kim. Conormal derivative problems for stationary Stokes system in Sobolev spaces. Discrete Contin. Dyn. Syst., 38(5):2349-2374, 2018. https://doi.org/10.3934/dcds.2018097
  4. Jongkeun Choi, Hongjie Dong, and Doyoon Kim. Green functions of conormal derivative problems for Stokes system. J. Math. Fluid Mech., 20(4):1745-1769, 2018. https://doi.org/10.1007/s00021-018-0387-0
  5. Jongkeun Choi and Seick Kim. Green's function for second order parabolic systems with Neumann boundary condition. J. Differential Equations, 254(7):2834-2860, 2013. https://doi.org/10.1016/j.jde.2013.01.003
  6. Jongkeun Choi and Seick Kim. Neumann functions for second order elliptic systems with measurable coefficients. Trans. Amer. Math. Soc., 365(12):6283-6307, 2013. https://doi.org/10.1090/S0002-9947-2013-05886-2
  7. Jongkeun Choi and Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Commun. Pure Appl. Anal., 16(6):1989-2022, 2017. https://doi.org/10.3934/cpaa.2017098
  8. Mariano Giaquinta. Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, 1993.
  9. Steve Hofmann and Seick Kim. The Green function estimates for strongly elliptic systems of second order. Manuscripta math., 124(2):139-172, 2007. https://doi.org/10.1007/s00229-007-0107-1
  10. Stanislav Kracmar and Jiri Neustupa. A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions. Nonlinear Anal., 47(6):4169-4180, 2001. https://doi.org/10.1016/S0362-546X(01)00534-X
  11. Vladimir Gilelevich Maz'ya and Jurgen Rossmann. Pointwise estimates for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone. Math. Nachr., 278(2005):1766-1810, 2005.  https://doi.org/10.1002/mana.200410340