Global regularity for 3D magneto-hydrodynamics equations with only horizontal dissipation

Abstract

We investigate the Cauchy problem for the 3D magnetohydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.

Keywords

Magneto-hydrodynamics (MHD) equations / degenerate dissipation / global solution / Green's function method / frequency decomposition method

References

[1] Abidi H, Hmidi T, Keraani S. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete Contin Dyn Syst, 2007, 29(3): 737–756

[2] Adams R A, Fournier J J F. Sobolev Spaces. 2nd. Pure Appl Math. Amsterdam: Elsevier/Academic Press, 2003

[3] Cao C S, Dipendra R, Wu J H. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J Diérential Equations, 2013, 254(7): 2661–2681

[4] Cao C S, Wu J H. Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv Math, 2011, 226(2): 1803–1822

[5] Chen J, Li Y C, Wang W K. Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion. J Diérential Equations, 2016, 260(5): 4657–4682

[6] Danchin R. Axisymmetric incompressible ows with bounded vorticity. Russian Math Surveys, 2007, 62(3): 475–496

[7] Duvaut D, Lions J L. Inéquations en thermoélasticité et magnétohydrodynamique. Arch Ration Mech Anal, 1972, 46: 241–279

[8] Hmidi T, Keraani S, Rousset F. Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data. Ann Inst H Poincaré Anal Non Linéaire, 2010, 27(5): 1227–1246

[9] Hmidi T, Rousset F. Global well-posedness for the Euler-Boussinesq system with axisymmetric data. J Funct Anal, 2011, 260(3): 745–796

[10] Jiu Q S, Liu J T. Global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Discrete Contin Dyn Syst, 2015, 35(1): 301–322

[11] Ladyzhenskaya O A. Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. Zap Nauchn Sem Leningrad Otdel Mat Inst Steklov (LOMI), 1968, 7: 155–177

[12] Lei Z. On axially symmetric incompressible magnetohydrodynamics in three dimension. J Diérential Equations, 2015, 259(7): 3202–3215

[13] Lei Z, Zhou Y. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin Dyn Syst, 2009, 25(2): 575–583

[14] Leonardi S, Málek J, Necas J, Pokorný M. On axially symmetric ows in R3: Z Anal Anwend, 1999, 18(3): 639–649

[15] Li T, Chen Y M. Nonlinear Evolution Equations. Beijing: Science Press, 1989 (in Chinese)

[16] Lin F H, Xu L, Zhang P. Global small solutions of 2-D incompressible MHD system. J Diérential Equations, 2015, 259(10): 5440–5485

[17] Miao C X, Zheng X X. On the global well-posedness for the Boussinesq system with horizontal dissipation. Comm Math Phys, 2013, 321(1): 33–67

[18] Miao C X, Zheng X X. Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity. J Math Pures Appl, 2014, 101(6): 842–872

[19] Ren X X, Wu J H, Xiang Z Y, Zhang Z F. Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J Funct Anal, 2014, 267(2): 503–541

[20] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36(5): 635–664

[21] Ukhovskii M R, Yudovich V I. Axially symmetric ows of ideal and viscous uids filling the whole space. J Appl Math Mech, 1968, 32: 52–61

[22] Zhang T. Global solutions to the 2D viscous, non-resistive MHD systems with large background magnetic field. J Diérential Equations, 2016, 260(6): 5450–5480

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature