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