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

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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

Abstract
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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 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.

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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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