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Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data
1. | Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China, China |
References:
[1] |
H. Abidi and T. Hmidi, Résultats d'existence dans des espaces critiques pour le système de la MHD inhomogène, (French) [Existence in critical spaces for the inhomogeneous MHD system], Ann. Math. Blaise Pascal, 14 (2007), 103-148.
doi: 10.5802/ambp.230. |
[2] |
H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.
doi: 10.1017/S0308210506001181. |
[3] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[4] |
C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[5] |
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. |
[6] |
J. Y. Chemin, Perfect Incompressible Fluids, The Clarendon Press, Oxford University Press, New York, 1998. |
[7] |
J. Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $\mathbfR^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.
doi: 10.1016/j.anihpc.2007.05.008. |
[8] |
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. |
[9] |
Q. Chen, Z. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.
doi: 10.1002/mma.1338. |
[10] |
R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[11] |
R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334.
doi: 10.1017/S030821050000295X. |
[12] |
R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386. |
[13] |
B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394. |
[14] |
G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[15] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[16] |
J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452. |
[17] |
J. F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for The Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[18] |
G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity, J. Funct. Anal., 267 (2014), 1488-1539.
doi: 10.1016/j.jfa.2014.06.002. |
[19] |
C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 238 (2007), 1-17.
doi: 10.1016/j.jde.2007.03.023. |
[20] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[21] |
C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.
doi: 10.1016/j.jfa.2005.06.009. |
[22] |
X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.
doi: 10.1016/j.jde.2012.08.029. |
[23] |
L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, $2^{nd}$ edition, Butterworth-Heinemann, U.K., 1999. |
[24] |
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. |
[25] |
P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996. |
[26] |
R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers Group, Dordrecht, 1990.
doi: 10.1007/978-94-015-7883-7. |
[27] |
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. |
[28] |
M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[29] |
J. Peetre, New Thoughts on Besov Spaces, Duke University, Durham, N.C., 1976. |
[30] |
R. V. Polovin and V. P. Demutskiĭ, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990. |
[31] |
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. |
[32] |
C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys., 56 (2015), 091512, 18 pp.
doi: 10.1063/1.4931467. |
[33] |
Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199.
doi: 10.1007/s00033-009-0023-1. |
show all references
References:
[1] |
H. Abidi and T. Hmidi, Résultats d'existence dans des espaces critiques pour le système de la MHD inhomogène, (French) [Existence in critical spaces for the inhomogeneous MHD system], Ann. Math. Blaise Pascal, 14 (2007), 103-148.
doi: 10.5802/ambp.230. |
[2] |
H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.
doi: 10.1017/S0308210506001181. |
[3] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[4] |
C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[5] |
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. |
[6] |
J. Y. Chemin, Perfect Incompressible Fluids, The Clarendon Press, Oxford University Press, New York, 1998. |
[7] |
J. Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $\mathbfR^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.
doi: 10.1016/j.anihpc.2007.05.008. |
[8] |
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. |
[9] |
Q. Chen, Z. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.
doi: 10.1002/mma.1338. |
[10] |
R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[11] |
R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334.
doi: 10.1017/S030821050000295X. |
[12] |
R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386. |
[13] |
B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394. |
[14] |
G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[15] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[16] |
J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452. |
[17] |
J. F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for The Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[18] |
G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity, J. Funct. Anal., 267 (2014), 1488-1539.
doi: 10.1016/j.jfa.2014.06.002. |
[19] |
C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 238 (2007), 1-17.
doi: 10.1016/j.jde.2007.03.023. |
[20] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[21] |
C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.
doi: 10.1016/j.jfa.2005.06.009. |
[22] |
X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.
doi: 10.1016/j.jde.2012.08.029. |
[23] |
L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, $2^{nd}$ edition, Butterworth-Heinemann, U.K., 1999. |
[24] |
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. |
[25] |
P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996. |
[26] |
R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers Group, Dordrecht, 1990.
doi: 10.1007/978-94-015-7883-7. |
[27] |
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. |
[28] |
M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[29] |
J. Peetre, New Thoughts on Besov Spaces, Duke University, Durham, N.C., 1976. |
[30] |
R. V. Polovin and V. P. Demutskiĭ, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990. |
[31] |
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. |
[32] |
C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys., 56 (2015), 091512, 18 pp.
doi: 10.1063/1.4931467. |
[33] |
Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199.
doi: 10.1007/s00033-009-0023-1. |
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