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July  2014, 13(4): 1553-1561. doi: 10.3934/cpaa.2014.13.1553

Global existence of strong solutions to incompressible MHD

1. 

The Institute of Mathematical Sciences, University of Science and Technology of China, Anhui, 230026, China

2. 

The Institute of Mathematical Sciences, The Chinese University of Hong Kong

Received  September 2013 Revised  November 2013 Published  February 2014

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.
Citation: Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
References:
[1]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics,, Addison–Wesley, (1965).   Google Scholar

[2]

L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media,, 2nd ed., (1984).   Google Scholar

[3]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 635.  doi: 10.1002/cpa.3160360506.  Google Scholar

[4]

G. Duvaut and J. L. Lions, Inequations en thermoelasticite et magnetohydrodynamique,, \emph{Ach.Rational Mech. Anal.}, 46 (1972), 241.   Google Scholar

[5]

J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation,, \emph{Adv. Differential Equations}, 2 (1997), 427.   Google Scholar

[6]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1996).   Google Scholar

[7]

P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models,, Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar

[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,, \emph{Arch. Rational Mech. Anal.}, 197 (2010), 203.  doi: 10.1007/s00205-010-0295-9.  Google Scholar

[9]

X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible Magnetohydrodynamic flows,, \emph{Commun. Math. Phys.}, 283 (2008), 255.  doi: 10.1007/s00220-008-0497-2.  Google Scholar

[10]

J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, \emph{Nonlinear Analysis}, 69 (2008), 3637.  doi: 10.1016/j.na.2007.10.005.  Google Scholar

[11]

E. Feireisl, Dynamics of viscous compressible fluids,, Oxford Lecture Series in Mathematics and its Applications, (2004).   Google Scholar

[12]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, \emph{J. Math. Fluid Mech.}, 3 (2001), 358.  doi: 10.1007/PL00000976.  Google Scholar

[13]

B. Ducomet and E. Feireisl, The equation of Magnetohydrodynamics: on the interaction between matter and ration in the evolution of gaseous stars,, \emph{Commun. Math. Phys.}, 266 (2006), 595.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[14]

Q. Chen, Z. Tan and Y. J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations,, \emph{Math. Methods Appl. Sci.}, 34 (2011), 94.  doi: 10.1002/mma.1338.  Google Scholar

[15]

H. W. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum,, \emph{Comput. Math. Appl.}, 61 (2011), 2742.  doi: 10.1016/j.camwa.2011.03.033.  Google Scholar

[16]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system,, \emph{J. Differential Equations}, 254 (2013), 511.  doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2007.10.001.  Google Scholar

[18]

X. L. Li, N. Su, and D. H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data,, \emph{J. Hyperbolic Differ. Equ.}, 8 (2011), 415.  doi: 10.1142/S0219891611002457.  Google Scholar

[19]

X. L. Li and D. H. Wang, Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows,, \emph{J. Differential Equations}, 251 (2011), 1580.  doi: 10.1016/j.jde.2011.06.004.  Google Scholar

[20]

W. Von Wahl, Estimating $\nabla u$ by $\text{div} u$ and $\text{curl}u$,, \emph{Math. Methods Appl. Sci.}, 15 (1992), 123.  doi: 10.1002/mma.1670150206.  Google Scholar

[21]

Y. Zhou, Remarks on regularities for the 3D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 881.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

[22]

Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, \emph{Internat. J. Non-Linear Mech.}, 41 (2006), 1174.  doi: 10.1016/j.ijnonlinmec.2006.12.001.  Google Scholar

[23]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 24 (2007), 491.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

show all references

References:
[1]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics,, Addison–Wesley, (1965).   Google Scholar

[2]

L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media,, 2nd ed., (1984).   Google Scholar

[3]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 635.  doi: 10.1002/cpa.3160360506.  Google Scholar

[4]

G. Duvaut and J. L. Lions, Inequations en thermoelasticite et magnetohydrodynamique,, \emph{Ach.Rational Mech. Anal.}, 46 (1972), 241.   Google Scholar

[5]

J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation,, \emph{Adv. Differential Equations}, 2 (1997), 427.   Google Scholar

[6]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1996).   Google Scholar

[7]

P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models,, Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar

[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,, \emph{Arch. Rational Mech. Anal.}, 197 (2010), 203.  doi: 10.1007/s00205-010-0295-9.  Google Scholar

[9]

X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible Magnetohydrodynamic flows,, \emph{Commun. Math. Phys.}, 283 (2008), 255.  doi: 10.1007/s00220-008-0497-2.  Google Scholar

[10]

J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, \emph{Nonlinear Analysis}, 69 (2008), 3637.  doi: 10.1016/j.na.2007.10.005.  Google Scholar

[11]

E. Feireisl, Dynamics of viscous compressible fluids,, Oxford Lecture Series in Mathematics and its Applications, (2004).   Google Scholar

[12]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, \emph{J. Math. Fluid Mech.}, 3 (2001), 358.  doi: 10.1007/PL00000976.  Google Scholar

[13]

B. Ducomet and E. Feireisl, The equation of Magnetohydrodynamics: on the interaction between matter and ration in the evolution of gaseous stars,, \emph{Commun. Math. Phys.}, 266 (2006), 595.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[14]

Q. Chen, Z. Tan and Y. J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations,, \emph{Math. Methods Appl. Sci.}, 34 (2011), 94.  doi: 10.1002/mma.1338.  Google Scholar

[15]

H. W. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum,, \emph{Comput. Math. Appl.}, 61 (2011), 2742.  doi: 10.1016/j.camwa.2011.03.033.  Google Scholar

[16]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system,, \emph{J. Differential Equations}, 254 (2013), 511.  doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2007.10.001.  Google Scholar

[18]

X. L. Li, N. Su, and D. H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data,, \emph{J. Hyperbolic Differ. Equ.}, 8 (2011), 415.  doi: 10.1142/S0219891611002457.  Google Scholar

[19]

X. L. Li and D. H. Wang, Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows,, \emph{J. Differential Equations}, 251 (2011), 1580.  doi: 10.1016/j.jde.2011.06.004.  Google Scholar

[20]

W. Von Wahl, Estimating $\nabla u$ by $\text{div} u$ and $\text{curl}u$,, \emph{Math. Methods Appl. Sci.}, 15 (1992), 123.  doi: 10.1002/mma.1670150206.  Google Scholar

[21]

Y. Zhou, Remarks on regularities for the 3D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 881.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

[22]

Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, \emph{Internat. J. Non-Linear Mech.}, 41 (2006), 1174.  doi: 10.1016/j.ijnonlinmec.2006.12.001.  Google Scholar

[23]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 24 (2007), 491.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

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