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January  2015, 35(1): 301-322. doi: 10.3934/dcds.2015.35.301

Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion

1. 

School of Mathematical Sciences & Beijing Center of Mathematics and Information Sciences, Capital Normal University, Beijing, 100048, China

2. 

The Institute of Mathematical Sciences, The Chinese University of Hong Kong, China

Received  January 2014 Revised  July 2014 Published  August 2014

Whether or not classical solutions of the 3D incompressible MHD equations with full dissipation and magnetic diffusion can develop finite-time singularities is a long standing open problem of fluid dynamics and PDE theory. In this paper, we investigate the Cauchy problem for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. We get a unique global smooth solution under the assumption that $u_\theta$ and $b_r$ are trivial. In absence of some viscosities, there is no smoothing effect on the derivatives of that direction. However, we take full advantage of the structures of MHD system to make up this shortcoming.
Citation: Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301
References:
[1]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455. doi: 10.1007/s002200050067.

[2]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[3]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.

[4]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317.

[5]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335. doi: 10.1051/m2an:2000143.

[6]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[7]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Funct. Anal., 260 (2011), 745-796. doi: 10.1016/j.jfa.2010.10.012.

[8]

T. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057.

[9]

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

[10]

Z. Lei, On axially symmetric incompressible Magnetohydrodynamics in three dimensions, preprint, arXiv:1212.5968v2.

[11]

S. Leonardi, J. Malek, J. Necas and M. Pokorny, On axially symmetric flows in $R^{3}$, Z. Anal. Anwendungen, 18 (1999), 639-649. doi: 10.4171/ZAA/903.

[12]

F.-H. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, preprint, arXiv:1302.5877v2.

[13]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67. doi: 10.1007/s00220-013-1721-2.

[14]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[15]

M. R. Ukhovskii and V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0.

[16]

J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265. doi: 10.1007/BF02788146.

show all references

References:
[1]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455. doi: 10.1007/s002200050067.

[2]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[3]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.

[4]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317.

[5]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335. doi: 10.1051/m2an:2000143.

[6]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[7]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Funct. Anal., 260 (2011), 745-796. doi: 10.1016/j.jfa.2010.10.012.

[8]

T. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057.

[9]

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

[10]

Z. Lei, On axially symmetric incompressible Magnetohydrodynamics in three dimensions, preprint, arXiv:1212.5968v2.

[11]

S. Leonardi, J. Malek, J. Necas and M. Pokorny, On axially symmetric flows in $R^{3}$, Z. Anal. Anwendungen, 18 (1999), 639-649. doi: 10.4171/ZAA/903.

[12]

F.-H. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, preprint, arXiv:1302.5877v2.

[13]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67. doi: 10.1007/s00220-013-1721-2.

[14]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[15]

M. R. Ukhovskii and V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0.

[16]

J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265. doi: 10.1007/BF02788146.

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