Frontiers of Mathematics in China

, Volume 14, Issue 1, pp 149–175 | Cite as

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

  • Yutong Wang
  • Weike WangEmail author
Research Article


We investigate the Cauchy problem for the 3D magneto-hydrodynamics 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.


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


35A01 35A22 35A25 


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This work was supported in part by the National Natural Science Foundation of China (Grant No. 11771284).


  1. 1.
    Abidi H, Hmidi T, Keraani S. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete Contin Dyn Syst, 2007, 29(3): 737–756MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams R A, Fournier J J F. Sobolev Spaces. 2nd. Pure Appl Math. Amsterdam: Elsevier/Academic Press, 2003Google Scholar
  3. 3.
    Cao C S, Dipendra R, Wu J H. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J Differential Equations, 2013, 254(7): 2661–2681MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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): 1803–1822MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen J, Li Y C, Wang W K. Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion. J Differential Equations, 2016, 260(5): 4657–4682MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Danchin R. Axisymmetric incompressible flows with bounded vorticity. Russian Math Surveys, 2007, 62(3): 475–496MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duvaut D, Lions J L. Inéquations en thermoélasticité et magnétohydrodynamique. Arch Ration Mech Anal, 1972, 46: 241–279CrossRefzbMATHGoogle Scholar
  8. 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): 1227–1246MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hmidi T, Rousset F. Global well-posedness for the Euler-Boussinesq system with axisymmetric data. J Funct Anal, 2011, 260(3): 745–796MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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): 301–322MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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: 155–177MathSciNetGoogle Scholar
  12. 12.
    Lei Z. On axially symmetric incompressible magnetohydrodynamics in three dimension. J Differential Equations, 2015, 259(7): 3202–3215MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lei Z, Zhou Y. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin Dyn Syst, 2009, 25(2): 575–583MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Leonardi S, Málek J, Necas J, Pokorný M. On axially symmetric flows in R3. Z Anal Anwend, 1999, 18(3): 639–649CrossRefzbMATHGoogle Scholar
  15. 15.
    Li T, Chen Y M. Nonlinear Evolution Equations. Beijing: Science Press, 1989 (in Chinese)Google Scholar
  16. 16.
    Lin F H, Xu L, Zhang P. Global small solutions of 2-D incompressible MHD system. J Differential Equations, 2015, 259(10): 5440–5485MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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): 33–67MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miao C X, Zheng X X. Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity. J Math Pures Appl, 2014, 101(6): 842–872MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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): 503–541MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36(5): 635–664MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ukhovskii M R, Yudovich V I. Axially symmetric flows of ideal and viscous fluids filling the whole space. J Appl Math Mech, 1968, 32: 52–61MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang T. Global solutions to the 2D viscous, non-resistive MHD systems with large background magnetic field. J Differential Equations, 2016, 260(6): 5450–5480MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiChina

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