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On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity
Global existence of strong solutions to incompressible MHD
1. | The Institute of Mathematical Sciences, University of Science and Technology of China, Anhui, 230026 |
2. | The Institute of Mathematical Sciences, The Chinese University of Hong Kong |
References:
[1] |
A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison–Wesley, Reading, MA, 1965. |
[2] |
L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon, New York, 1984. |
[3] |
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. |
[4] |
G. Duvaut and J. L. Lions, Inequations en thermoelasticite et magnetohydrodynamique, Ach.Rational Mech. Anal., 46 (1972), 241-279. |
[5] |
J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452. |
[6] |
P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. |
[7] |
P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. |
[8] |
X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible Magnetohydrodynamic flows, Arch. Rational Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
[9] |
X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible Magnetohydrodynamic flows, Commun. Math. Phys., 283 (2008), 255-284.
doi: 10.1007/s00220-008-0497-2. |
[10] |
J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Analysis, 69 (2008), 3637-3660.
doi: 10.1016/j.na.2007.10.005. |
[11] |
E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. |
[12] |
E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[13] |
B. Ducomet and E. Feireisl, The equation of Magnetohydrodynamics: on the interaction between matter and ration in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
[14] |
Q. Chen, Z. Tan and Y. J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.
doi: 10.1002/mma.1338. |
[15] |
H. W. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753.
doi: 10.1016/j.camwa.2011.03.033. |
[16] |
X. D. 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. |
[17] |
J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.
doi: 10.1016/j.nonrwa.2007.10.001. |
[18] |
X. L. Li, N. Su, and D. H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436.
doi: 10.1142/S0219891611002457. |
[19] |
X. L. Li and D. H. Wang, Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows, J. Differential Equations, 251 (2011), 1580-1615.
doi: 10.1016/j.jde.2011.06.004. |
[20] |
W. Von Wahl, Estimating $\nabla u$ by $\text{div} u$ and $\text{curl}u$, Math. Methods Appl. Sci., 15 (1992), 123-143.
doi: 10.1002/mma.1670150206. |
[21] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.
doi: 10.3934/dcds.2005.12.881. |
[22] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: 10.1016/j.ijnonlinmec.2006.12.001. |
[23] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.
doi: 10.1016/j.anihpc.2006.03.014. |
show all references
References:
[1] |
A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison–Wesley, Reading, MA, 1965. |
[2] |
L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon, New York, 1984. |
[3] |
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. |
[4] |
G. Duvaut and J. L. Lions, Inequations en thermoelasticite et magnetohydrodynamique, Ach.Rational Mech. Anal., 46 (1972), 241-279. |
[5] |
J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452. |
[6] |
P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. |
[7] |
P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. |
[8] |
X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible Magnetohydrodynamic flows, Arch. Rational Mech. Anal., 197 (2010), 203-238.
doi: 10.1007/s00205-010-0295-9. |
[9] |
X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible Magnetohydrodynamic flows, Commun. Math. Phys., 283 (2008), 255-284.
doi: 10.1007/s00220-008-0497-2. |
[10] |
J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Analysis, 69 (2008), 3637-3660.
doi: 10.1016/j.na.2007.10.005. |
[11] |
E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. |
[12] |
E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[13] |
B. Ducomet and E. Feireisl, The equation of Magnetohydrodynamics: on the interaction between matter and ration in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
[14] |
Q. Chen, Z. Tan and Y. J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.
doi: 10.1002/mma.1338. |
[15] |
H. W. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753.
doi: 10.1016/j.camwa.2011.03.033. |
[16] |
X. D. 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. |
[17] |
J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.
doi: 10.1016/j.nonrwa.2007.10.001. |
[18] |
X. L. Li, N. Su, and D. H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436.
doi: 10.1142/S0219891611002457. |
[19] |
X. L. Li and D. H. Wang, Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows, J. Differential Equations, 251 (2011), 1580-1615.
doi: 10.1016/j.jde.2011.06.004. |
[20] |
W. Von Wahl, Estimating $\nabla u$ by $\text{div} u$ and $\text{curl}u$, Math. Methods Appl. Sci., 15 (1992), 123-143.
doi: 10.1002/mma.1670150206. |
[21] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.
doi: 10.3934/dcds.2005.12.881. |
[22] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: 10.1016/j.ijnonlinmec.2006.12.001. |
[23] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.
doi: 10.1016/j.anihpc.2006.03.014. |
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