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

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

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 $\varepsilon_0.$
Citation: Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5801-5815. doi: 10.3934/dcds.2016055
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

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