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].