Global existence of strong solutions to incompressible MHD

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May 2014, 13(3): 1337-1345. doi: 10.3934/cpaa.2014.13.1337

Global existence of strong solutions to incompressible MHD

Huajun Gong ^1, and Jinkai Li ^2,

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

Received September 2013 Revised November 2013 Published December 2013

Abstract
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We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\|\sqrt\rho_0u_0\|_{L^2(\Omega)}^2+\|H_0\|_{L^2(\Omega)}^2$ and $\|\nabla u_0\|_{L^2(\Omega)}^2+\|\nabla H_0\|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.

Keywords: Incompressible MHD, strong solutions., global existence and uniqueness.

Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D0.

Citation: Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

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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