# American Institute of Mathematical Sciences

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

 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.
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:
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##### 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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