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

Jishan Fan; Fucai Li; Gen Nakamura

doi:10.3934/cpaa.2014.13.1481

Article Contents

2014, Volume 13, Issue 4: 1481-1490. Doi: 10.3934/cpaa.2014.13.1481

This issue Previous Article Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation Next Article Multiple solutions for a class of nonlinear Neumann eigenvalue problems

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

Received: May 31, 2013

Revised: December 31, 2013

Published: June 2015

Abstract Related Papers Cited by

Abstract

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,
vaccum,
global existence,
uniform regularity.

Mathematics Subject Classification: Primary: 76W05; Secondary: 76D03, 76D09.

Citation:

Related Papers

Cited by

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, Acta Appl. Math., published online. 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.