ORIGINAL ARTICLE
Green functions for stationary Stokes systems with conormal derivative boundary condition in two dimensions
Jongkeun Choi,
Doyoon Kim,
Jongkeun Choi
Department of Mathematics Education, Pusan National University, Busan, Republic of Korea
Search for more papers by this authorCorresponding Author
Doyoon Kim
Department of Mathematics, Korea University, Seoul, Republic of Korea
Correspondence
Doyoon Kim, Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea.
Email: [email protected]
Search for more papers by this authorJongkeun Choi,
Doyoon Kim,
Jongkeun Choi
Department of Mathematics Education, Pusan National University, Busan, Republic of Korea
Search for more papers by this authorCorresponding Author
Doyoon Kim
Department of Mathematics, Korea University, Seoul, Republic of Korea
Correspondence
Doyoon Kim, Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea.
Email: [email protected]
Search for more papers by this authorFirst published: 15 December 2023
Abstract
We construct Green functions of conormal derivative problems for the stationary Stokes system with measurable coefficients in a two-dimensional Reifenberg flat domain.
REFERENCES
- 1H. Abidi, G. Gui, and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math. 64 (2011), no. 6, 832–881.
- 2M. Beneš and P. Kučera, Solutions to the Navier-Stokes equations with mixed boundary conditions in two-dimensional bounded domains, Math. Nachr. 289 (2016), no. 2-3, 194–212.
- 3S. Cho, H. Dong, and S. Kim, Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains, Potential Anal. 36 (2012), no. 2, 339–372.
- 4J. Choi and H. Dong, Gradient estimates for Stokes systems with Dini mean oscillation coefficients, J. Differential Equations 266 (2019), no. 8, 4451–4509.
- 5J. Choi and H. Dong, Green functions for the pressure of Stokes systems, Int. Math. Res. Not. IMRN 2021 (2021), no. 3, 1699–1759.
- 6J. Choi, H. Dong, and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst. 38 (2018), no. 5, 2349–2374.
- 7J. Choi, H. Dong, and D. Kim, Green functions of conormal derivative problems for Stokes system, J. Math. Fluid Mech. 20 (2018), no. 4, 1745–1769.
- 8J. Choi, H. Dong, and Z. Li., Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain, Appl. Math. Optim. 83 (2021), no. 3, 1547–1583.
- 9J. Choi, H. Dong, and L. Xu, Gradient estimates for stokes and Navier-Stokes systems with piecewise DMO coefficients, SIAM J. Math. Anal. 54 (2022), no. 3, 3609–3635.
- 10J. Choi and D. Kim, Estimates for Green functions of Stokes systems in two dimensional domains, J. Math. Anal. Appl. 471 (2019), no. 1-2, 102–125.
- 11J. Choi and S. Kim, Green's function for second order parabolic systems with Neumann boundary condition, J. Differential Equations 254 (2013), no. 7, 2834–2860.
- 12J. Choi and S. Kim, Neumann functions for second order elliptic systems with measurable coefficients, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6283–6307.
- 13J. Choi and S. Kim, Green's functions for elliptic and parabolic systems with Robin-type boundary conditions, J. Funct. Anal. 267 (2014), no. 9, 3205–3261.
- 14J. Choi and K.-A. Lee, The Green function for the Stokes system with measurable coefficients, Commun. Pure Appl. Anal. 16 (2017), no. 6, 1989–2022.
- 15J. Choi and M. Yang, Fundamental solutions for stationary Stokes systems with measurable coefficients, J. Differential Equations 263 (2017), no. 7, 3854–3893.
- 16G. Dolzmann and S. Müller, Estimates for Green's matrices of elliptic systems by theory, Manuscripta Math. 88 (1995), no. 2, 261–273.
- 17H. Dong and D. Kim, Weighted -estimates for stationary Stokes system with partially BMO coefficients, J. Differential Equations 264 (2018), no. 7, 4603–4649.
- 18H. Dong and S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3303–3323.
- 19M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983.
- 20M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342.
- 21S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math. 124 (2007), no. 2, 139–172.
- 22K. Kang and S. Kim, Global pointwise estimates for Green's matrix of second order elliptic systems, J. Differential Equations 249 (2010), no. 11, 2643–2662.
- 23S. Kračmar and J. Neustupa, A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions, Nonlinear Anal. 47 (2001), no. 6, 4169–4180.
- 24O. A. Ladyženskaja and V. A. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52 (1975), 52–109, 218–219.
- 25P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996.
- 26J. H. Masliyah, G. Neale, K. Malysa, and T. G. M. Van De Ven, Creeping flow over a composite sphere: Solid core with porous shell, Chem. Eng. Sci. 42 (1987), no. 2, 245–253.
- 27D. Mitrea and I. Mitrea, On the regularity of Green functions in Lipschitz domains, Comm. Partial Differ. Equations 36 (2011), no. 2, 304–327.
- 28K. A. Ott, S. Kim, and R. M. Brown, The Green function for the mixed problem for the linear Stokes system in domains in the plane, Math. Nachr. 288 (2015), no. 4, 452–464.
- 29J. L. Taylor, S. Kim, and R. M. Brown, The Green function for elliptic systems in two dimensions, Comm. Partial Differentix 38 (2013), no. 9, 1574–1600.
- 30J. Xiong and J. Bao, Sharp regularity for elliptic systems associated with transmission problems, Potential Anal. 39 (2013), no. 2, 169–194.
