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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996.
[11]	E.-H. Lieb and M. Loss, Analysis, 2nd ed., American Mathematical Society, Providence, RI, 2001.
[12]	A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.
[13]	T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: pii/S0022123685710129.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996.
[11]	E.-H. Lieb and M. Loss, Analysis, 2nd ed., American Mathematical Society, Providence, RI, 2001.
[12]	A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.
[13]	T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: pii/S0022123685710129.
[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.

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