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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 |
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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