Global Large Strong Solutions of Radially Symmetric Compressible MHD Equations in 2D Discs

This paper is devoted to the study of the Dirichlet problem for the compressible magnetohydrodynamic system with density-dependent viscosities \(\mu =const>0,\lambda =\rho ^\beta \) which was first introduced by Vaigant-Kazhikhov [18] in 1995. By assuming the endpoint case \(\beta =1\) in the radially spherical symmetric setting, we establish the global existence to strong solution of the two-dimensional system for any large initial data. This also improves the previous work of Huang-Yan [10] where they proved the similar result for \(\beta >1\). Our main idea is to utilize the geometric structure of a 2D spherically symmetric disc and the Sobolev critical embedding inequality of spherically symmetric functions in 2D domains, as well as a refined estimate of the upper bound of the density.