GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY*

doi:10.1007/s10473-023-0316-z

Abstract

Abstract: This paper studies the global regularity of 2D incompressible anisotropic magneto-micropolar fluid equations with partial viscosity. Ma [22] (Ma L. Nonlinear Anal: Real World Appl, 2018, 40: 95-129) examined the global regularity of the 2D incompressible magneto-micropolar fluid system for 21 anisotropic partial viscosity cases. He proved the global existence of a classical solution for some cases and established the conditional global regularity for some other cases. In this paper, we also investigate the global regularity of 12 cases in [22] and some other new partial viscosity cases. The global regularity is established by providing new regular conditions. Our work improves some results in [22] in this sense of weaker regular criteria.

Key words: magneto-micropolar fluid equations, partial dissipation, global regularity, regular criteria

Hongxia Lin, Sen Liu, Heng Zhang, Ru Bai. GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY^*[J].Acta mathematica scientia,Series B, 2023, 43(3): 1275-1300.

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References

[1] Broadman N, Lin H, Wu J.Stabilization of a background magnetic field on a 2 dimensional magnetohy- drodynamic flow. SIAM J Math Anal, 2020, 52(5): 5001-5035
[2] Cao C, Regmi D, Wu J.The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J Differ Equ, 2013, 254: 2661-2681
[3] Cao C, Wu J.Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv in Math, 2011, 226: 1803-1822
[4] Chen Q, Miao C.Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equation in R3. J Differ Equ, 2007, 239: 251-271
[5] Cheng J, Liu Y.Global regularity of the 2D magnetic micropolar fluid flows with mixed partial viscosity. Compu Math Appl, 2015, 70: 66-72
[6] Du L, Zhou D.Global well-posedness of 2D magnetohydrodynamics flows with partial dissipation and magnetic diffusion. SIAM J Math Anal, 2015, 47: 1562-1587
[7] Duvaut G, Lions J.Inequations enthormolasticit et magnetohydrodynamique. Arch Rational Mech Anal, 1972, 46: 241-279
[8] Eringen A.Theory of micropolar fluids. J Math Mech, 1966, 16: 1-18
[9] Fan F, Ozawa T.Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinet Relat Models, 2014, 7: 45-56
[10] Gala G, Ragusa A, Ye Z.An improved blow-up criterion for smooth solutions of the two-dimensional MHD equations. Math Meth Appl Sci, 2017, 40: 279-285
[11] Gala S, Ragusa M, Zhang Z.A regularity criterion in terms of pressure for the 3D viscous MHD equations. Bull Malays Math Sci Soc, 2017, 40: 1677-1690
[12] He C, Xin Z.On the regularity of weak solutions to the magnetohydrodynamic equations. J Differ Equ, 2005, 213: 235-254
[13] Hu X, Lin F.Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity. arXiv:1405.0082
[14] Jiu Q, Niu D, Wu J, et al.The 2D magnetohydrodynamic equations with magnetic diffusion. Nonlinearity, 2015, 28(11): 3935-3955
[15] Li J, Zheng X.The well-posedness of the incompressible Magnetohydro Dynamic equations in the framework of Fourier-Herz space. J Differ Equ, 2017, 263: 3419-3459
[16] Lin H, Du L.Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions. Nonlinearity, 2013, 26: 219-239
[17] Lin F, Xu L, Zhang P.Global small solutions to 2D incompressible MHD system. J Differ Equ, 2015, 259: 5440-5485
[18] Lin H, Ji R, Wu J, Yan L.Stability of perturbations near a background magnetic field of the 2D incom- pressible MHD equations with mixed partial dissipation. J Func Anal, 2020, 279(2): 108519
[19] Lei Z, Zhou Y.BKM's criterion and global weak solutions for magnetohydrodynamics with zero visccosity. Discrete Contin Dyn Syst, 2009, 25: 575-583
[20] Miao C, Yuan B.On the well-posedness of the Cauchy problem for an MHD system in Besov spaces. Math Methods Appl Sci, 2009, 32: 53-76
[21] Miao C, Yuan B, Zhang B.Well-posedness for the incompressible magneto-hydrodynamic system. Math Methods Appl Sci, 2007, 30: 961-976
[22] Ma L.On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity. Non- linear Anal: Real World Appl, 2018, 40: 95-129
[23] Rojas-Medar M, Boldrini J.Magneto-micropolar fluid motionexistence of weak solution. Rev Mat Complut, 1998, 11: 443-460
[24] Regmi D, Wu J.Global regularity for the 2D magneto-micropolar equations with partial dissipation. J Math Study, 2016, 49: 169-194
[25] Ren X, Wu J, Xiang Z, Zhang Z.Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J Funct Anal, 2015, 267: 5440-5485
[26] Shang H, Gu C. Global regularity and decay estimates for 2D magneto-micropolar equations with partial dissipation. Z Angrew Math Phys, 2019, 70: Art 85
[27] Shang H, Zhao J.Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion. Nonlinear Anal, 2017, 150: 194-209
[28] Sermange M, Temam R.Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36: 635-664
[29] Wang B, Chen M. An improved pressure regularity criterion of magnetohydrodynamic equations in critical Besov spaces. Bound Value Probl, 2015, 2015: Art 66
[30] Wang K, Du Y.Stability of the two dimensional magnetohydrodynamic flows in R3. Discrete Contin Dyn Syst Ser B, 2012, 17: 1061-1073
[31] Wang Y, Wang K.Global well-posedness of the three dimensional magnetohydrodynamics equations. Non- linear Anal: RWA, 2014, 17: 245-251
[32] 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
[33] Yamazaki K.Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete Contin Dyn Syst, 2015, 35: 2193-2207
[34] Yu Y, Wu X, Tang Y. A magnetic regularity criterion for the 2D MHD equations with velocity dissipation. Boundary Value Problems, 2016, 2016: Art 113
[35] Zhang T.An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non- resistive MHD system. arXiv:1404.5681
[36] Zhou Y, Fan J.A regularity criterion for the 2D MHD system with zero magnetic diffusivity. J Math Anal Appl, 2011, 378: 169-172

GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY^*

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