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July  2014, 13(4): 1481-1490. doi: 10.3934/cpaa.2014.13.1481

Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum

Jishan Fan 1, , Fucai Li 2, and  Gen Nakamura 3,

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037
2. 

Department of Mathematics, Nanjing University, Nanjing 210093
3. 

Department of Mathematics, Inha University, Incheon 402-751, South Korea

Received  June 2013 Revised  January 2014 Published  February 2014

In this paper we establish the global existence of strong solution to the density-dependent incompressible magnetohydrodynamic equations with vaccum in a bounded domain in $R^2$. Furthermore, the limit as the heat conductivity coefficient tends to zero is also obtained.
Keywords: Density-dependent magnetohydrodynamic equations, uniform regularity., global existence, vaccum.
Mathematics Subject Classification: Primary: 76W05; Secondary: 76D03, 76D0.
Citation: Jishan Fan, Fucai Li, Gen Nakamura. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1481-1490. doi: 10.3934/cpaa.2014.13.1481
References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Pure and Appl. Math. (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

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J.-S. Fan and F.-C. Li, Uniform local well-posedness to the density-dependent Navier-Stokes-Maxwell system,, \emph{Acta Appl. Math.}, ().  doi: 10.1007/s10440-013-9857-9.  Google Scholar

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J.-F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006.  Google Scholar

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T. Li and T. Qin, Physics and Partial Differential Equations, Volume 1. Translated from the Chinese original by Yachun Li. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012.  Google Scholar

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P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

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E.-H. Lieb and M. Loss, Analysis, 2nd ed., American Mathematical Society, Providence, RI, 2001.  Google Scholar

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A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.  Google Scholar

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T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: pii/S0022123685710129.  Google Scholar

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H. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753. doi: 10.1016/j.camwa.2011.03.03.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Pure and Appl. Math. (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and its Applications, 22. North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar

[3]

H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/2Fs00021-009-0295-4.  Google Scholar

[4]

J.-S. Fan and F.-C. Li, Uniform local well-posedness to the density-dependent Navier-Stokes-Maxwell system,, \emph{Acta Appl. Math.}, ().  doi: 10.1007/s10440-013-9857-9.  Google Scholar

[5]

J.-F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006.  Google Scholar

[6]

S. Itoh, On the vanishing viscosity in the Cauchy problem for the equations of a nonhomogeneous incompressible fluid, Glasgow Math. J., 36 (1994), 123-129. doi: 10.1017/S0017089500030639.  Google Scholar

[7]

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.2012.08.02.  Google Scholar

[8]

M. L. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/2Fs00205-010-0357-z.  Google Scholar

[9]

T. Li and T. Qin, Physics and Partial Differential Equations, Volume 1. Translated from the Chinese original by Yachun Li. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012.  Google Scholar

[10]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[11]

E.-H. Lieb and M. Loss, Analysis, 2nd ed., American Mathematical Society, Providence, RI, 2001.  Google Scholar

[12]

A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.  Google Scholar

[13]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: pii/S0022123685710129.  Google Scholar

[14]

H. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753. doi: 10.1016/j.camwa.2011.03.03.  Google Scholar

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