Abstract
In this paper, we consider an initial-boundary value problem for the 2D incompressible magneto-micropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions. We establish the global well-posedness of strong solutions around the equilibrium (0, e1, 0).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abidi H, Zhang P. On the global solution of a 3-D MHD system with initial data near equilibrium. Comm Pure Appl Math, 2017, 70: 1509–1561
Berkovski B, Bashtovoy V. Magnetic Fluids and Applications Handbook. New York: Begell House, 1996
Cai Y, Lei Z. Global well-posedness of the incompressible magnetohydrodynamics. Arch Ration Mech Anal, 2018, 228: 969–993
Cao C, Regmi D, Wu J. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J Differential Equations, 2013, 254: 2661–2681
Cao C, Wu J. Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv Math, 2011, 226: 1803–1822
Cao C, Wu J, Yuan B. The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J Math Anal, 2014, 46: 588–602
Chen Q, Miao C. Global well-posedness for the micropolar fluid system in critical Besov spaces. J Differential Equations, 2012, 252: 2698–2724
Cheng J, Liu Y. Global regularity of the 2D magnetic micropolar fluid flows with mixed partial viscosity. Comput Math Appl, 2015, 70: 66–72
Deng W, Zhang P. Large time behavior of solutions to 3-D MHD system with initial data near equilibrium. Arch Ration Mech Anal, 2018, 230: 1017–1102
Dong B, Chen Z. Regularity criteria of weak solutions to the three-dimensional micropolar flows. J Math Phys, 2009, 50: 103525
Dong B, Chen Z. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete Contin Dyn Syst, 2009, 23: 765–784
Dong B, Li J, Wu J. Global well-posedness and large-time decay for the 2D micropolar equations. J Differential Equations, 2017, 262: 3488–3523
Dong B, Zhang Z. Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J Differential Equations, 2010, 249: 200–213
Du L, Zhou D. Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion. SIAM J Math Anal, 2015, 47: 1562–1589
Duvaut G, Lions J L. Inéquations en thermoélasticité et magnétohydrodynamique. Arch Ration Mech Anal, 1972, 46: 241–279
Eringen A C. Theory of micropolar fluids. J Math Mech, 1966, 16: 1–18
Galdi G, Rionero S. A note on the existence and uniqueness of solutions of micropolar fluid equations. Internat J Engrg Sci, 1977, 14: 105–108
Jiu Q, Niu D, Wu J, et al. The 2D magnetohydrodynamic equations with magnetic diffusion. Nonlinearity, 2015, 28: 3935–3955
Lei Z. On axially symmetric incompressible magnetohydrodynamics in three dimensions. J Differential Equations, 2015, 259: 3202–3215
Lin F, Xu L, Zhang P. Global small solutions to 2D incompressible MHD system. J Differential Equations, 2015, 259: 5440–5485
Lin F, Zhang P. Global small solutions to MHD type system (I): 3D case. Comm Pure Appl Math, 2014, 67: 531–580
Lin F, Zhang T. Global small solutions to a complex fluid model in 3D. Arch Ration Mech Anal, 2015, 216: 905–920
Pan R, Zhou Y, Zhu Y. Global classical solutions of three dimensional viscous MHD system without magnetic diffusion on periodic boxes. Arch Ration Mech Anal, 2018, 227: 637–662
Ren X, Wu J, Xiang Z, et al. Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J Funct Anal, 2015, 267: 5440–5485
Ren X, Xiang Z, Zhang Z. Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain. Nonlinearity, 2016, 29: 1257–1291
Ren X X, Xiang Z Y, Zhang Z F. Global existence and decay of smooth solutions for the 3-D MHD-type equations without magnetic diffusion. Sci China Math, 2016, 59: 1949–1974
Ren X X, Xiang Z Y, Zhang Z F. Low regularity well-posedness for the viscous surface wave equation. Sci China Math, 2019, https://doi.org/10.1007/s11425-018-9410-3
Rojas-Medar M A, Boldrini J L. Magneto-micropolar fluid motion: Existence of weak solutions. Rev Mat Complut, 1998, 11: 443–460
Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36: 635–664
Tan Z, Wang Y. Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems. SIAM J Math Anal, 2018, 50: 1432–1470
Wang Y, Wang Y. Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity. Math Methods Appl Sci, 2011, 34: 2125–2135
Wei R, Guo B, Li Y. Global existence and optimal convergence rates of solutions for 3D compressible magneto-micropolar fluid equations. J Differential Equations, 2017, 263: 2457–2480
Xiang Z, Yang H. On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative. Bound Value Probl, 2012, 2012: 139
Xu L, Zhang P. Global small solutions to three-dimensional incompressible magnetohydrodynamical system. SIAM J Math Anal, 2015, 47: 26–65
Xue L. Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math Methods Appl Sci, 2011, 34: 1760–1777
Yamazaki K. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete Contin Dyn Syst, 2015, 35: 2193–2207
Zhai Z, Zhang T. Global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system with non-equilibrium background magnetic field. J Differential Equations, 2016, 261: 3519–3550
Zhang T. An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system. ArXiv:1404.5681, 2014
Zhang T. Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field. J Differential Equations, 2016, 260: 5450–5480
Zhou Y, Zhu Y. Global classical solutions of 2D MHD system with only magnetic diffusion on periodic domain. J Math Phys, 2018, 59: 081505
Acknowledgements
The first author was supported by National Natural Science Foundation of China (Grant No. 11701049), the China Postdoctoral Science Foundation (Grant No. 2017M622989) and the Opening Fund of Geomathematics Key Laboratory of Sichuan Province (Grant No. scsxdz201707). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11571063 and 11771045). The authors are very grateful to the referees for their detailed comments and helpful suggestions, which greatly improved the manuscript, and to Professor Lili Du for suggesting this problem.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, H., Xiang, Z. Global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity. Sci. China Math. 63, 1285–1306 (2020). https://doi.org/10.1007/s11425-018-9427-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-018-9427-6
Keywords
- global well-posedness
- 2D magneto-micropolar fluid equations
- zero magnetic diffusion
- zero spin viscosity