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October  2016, 36(10): 5801-5815. doi: 10.3934/dcds.2016055

On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations

Gaocheng Yue 1, and  Chengkui Zhong 2,

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106
2. 

Department of Mathematics, Nanjing University, Nanjing 210093

Received  September 2015 Revised  April 2016 Published  July 2016

The present paper is devoted to the well-posedness issue of solutions to the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations with horizontal dissipation and horizontal magnetic diffusion. By means of anisotropic Littlewood-Paley analysis we prove the global well-posedness of solutions in the anisotropic Sobolev spaces of type $H^{0,s_0}(\mathbb{R}^3)$ with $s_0>\frac1{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} (\|u_n^h(0)\|_{H^{0,s_0}}^2+\|B_n^h(0)\|_{H^{0,s_0}}^2)\exp \Big\{C_1(\|u_0^3\|_{H^{0,s_0}}^4+\|B_0^3\|_{H^{0,s_0}}^4)\Big\}\leq\varepsilon_0, \end{align*} for some sufficiently small constant $\varepsilon_0.$
Keywords: anisotropic Littlewood-Paley theory, MHD equations, Well-posedness, Besov spaces..
Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 35Q3.
Citation: Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5801-5815. doi: 10.3934/dcds.2016055
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H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equ., 254 (2013), 2661-2681. doi: 10.1016/j.jde.2013.01.002.  Google Scholar

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J. Chemin, D. McCormick, J. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31. doi: 10.1016/j.aim.2015.09.004.  Google Scholar

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Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations, Comm. Math. Phys., 275 (2007), 861-872. doi: 10.1007/s00220-007-0319-y.  Google Scholar

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Q. Chen, C. Miao and Z. Zhang, On the Regularity Criterion of Weak Solution for the 3D Viscous Magneto-Hydrodynamics Equations, Comm. Math. Phys., 284 (2008), 919-930. doi: 10.1007/s00220-008-0545-y.  Google Scholar

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J. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566. doi: 10.1007/s00220-007-0236-0.  Google Scholar

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G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284. doi: 10.1016/j.aim.2010.03.022.  Google Scholar

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D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces, Rev. Mat. Iberoamericana, 15 (1999), 1-36. doi: 10.4171/RMI/248.  Google Scholar

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Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583. doi: 10.3934/dcds.2009.25.575.  Google Scholar

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F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differ. Equ., 259 (2015), 5440-5485. doi: 10.1016/j.jde.2015.06.034.  Google Scholar

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F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.  Google Scholar

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M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235. doi: 10.4171/RMI/420.  Google Scholar

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M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759. doi: 10.1007/s00220-011-1350-6.  Google Scholar

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[17]

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.  Google Scholar

[18]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Science, 12 (2002), 395-413. doi: 10.1007/s00332-002-0486-0.  Google Scholar

[19]

J. Wu, Regularity criteria for the generalized MHD equations, Commun. Partial Diff. Equ., 33 (2008), 285-306. doi: 10.1080/03605300701382530.  Google Scholar

[20]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656. doi: 10.1137/140985445.  Google Scholar

[21]

L. Xu and P. Zhang, Global small solutions to three-dimensional incompressible magnetohydrodynamical system, SIAM J. Math. Anal., 47 (2015), 26-65. doi: 10.1137/14095515X.  Google Scholar

[22]

T. Zhang, Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 295 (2010), 877-884. doi: 10.1007/s00220-010-1004-0.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equ., 254 (2013), 2661-2681. doi: 10.1016/j.jde.2013.01.002.  Google Scholar

[3]

J. Chemin, Localization in Fourier space and Navier-Stokes system Phase space analysis of Partial Differential Equations, Pubbl. Cert. Ric. Mat. Ennio de Gorg Scuola Norma. Sup. Pisa, I (2004), 53-135.  Google Scholar

[4]

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

[5]

J. Chemin, D. McCormick, J. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31. doi: 10.1016/j.aim.2015.09.004.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations, Comm. Math. Phys., 275 (2007), 861-872. doi: 10.1007/s00220-007-0319-y.  Google Scholar

[7]

Q. Chen, C. Miao and Z. Zhang, On the Regularity Criterion of Weak Solution for the 3D Viscous Magneto-Hydrodynamics Equations, Comm. Math. Phys., 284 (2008), 919-930. doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

J. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566. doi: 10.1007/s00220-007-0236-0.  Google Scholar

[9]

G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284. doi: 10.1016/j.aim.2010.03.022.  Google Scholar

[10]

D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces, Rev. Mat. Iberoamericana, 15 (1999), 1-36. doi: 10.4171/RMI/248.  Google Scholar

[11]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583. doi: 10.3934/dcds.2009.25.575.  Google Scholar

[12]

F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differ. Equ., 259 (2015), 5440-5485. doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[13]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.  Google Scholar

[14]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235. doi: 10.4171/RMI/420.  Google Scholar

[15]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759. doi: 10.1007/s00220-011-1350-6.  Google Scholar

[16]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[17]

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.  Google Scholar

[18]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Science, 12 (2002), 395-413. doi: 10.1007/s00332-002-0486-0.  Google Scholar

[19]

J. Wu, Regularity criteria for the generalized MHD equations, Commun. Partial Diff. Equ., 33 (2008), 285-306. doi: 10.1080/03605300701382530.  Google Scholar

[20]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656. doi: 10.1137/140985445.  Google Scholar

[21]

L. Xu and P. Zhang, Global small solutions to three-dimensional incompressible magnetohydrodynamical system, SIAM J. Math. Anal., 47 (2015), 26-65. doi: 10.1137/14095515X.  Google Scholar

[22]

T. Zhang, Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 295 (2010), 877-884. doi: 10.1007/s00220-010-1004-0.  Google Scholar

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