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November  2009, 8(6): 1841-1865. doi: 10.3934/cpaa.2009.8.1841

Global cylindrical solution to the compressible MHD equations in an exterior domain

Ming He 1, and  Jianwen Zhang 2,

1. 

School of Information Management, Jiangxi University of Finance & Economics, Nanchang 330013, China
2. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received  September 2008 Revised  March 2009 Published  August 2009

This paper is concerned with the global existence of cylindrical solution to an initial-boundary value problem for the magnetohydrodynamic equations in an exterior domain. The difficulty of the proof first lies in that the domain is unbounded and the coefficients tend to infinity as $x\to\infty$. Secondly, the additional nonlinear terms and nonlinear equations induced by magnetic field also make the problem more complicated than that for the compressible Navier-Stokes equations. To overcome such difficulties, we study approximate problems in bounded annular domains and assume that the heat conductivity satisfies certain physical growth condition. By virtue of the global (weighted) a priori estimates independent of the boundedness of the annular domain, letting the diameter of the annular domain go to infinity, we obtain the global existence theorem by the similar limit procedure as that in [23].
Keywords: global solution., exterior domain, cylindrical symme- try, a priori estimates, Magnetohydrodynamics (MHD).
Mathematics Subject Classification: Primary: 76W05, 76N10; Secondary: 35L6.
Citation: Ming He, Jianwen Zhang. Global cylindrical solution to the compressible MHD equations in an exterior domain. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1841-1865. doi: 10.3934/cpaa.2009.8.1841
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