Global well-posedness and large time asymptotic behavior of strong   solutions to the 2-D compressible magnetohydrodynamic equations with vacuum
  The authors study the Cauchy problem of the magnetohydrodynamic equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For two spatial dimensions, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) provided the smooth initial data are of small total energy, and obtain some a priori decay with rates (in large time) for the pressure, the spatial gradient of both the velocity field and the magnetic field. Moreover, for three spatial dimensions case, some similar decay rates are also obtained. 
Boqiang Lv, Xiaoding Shi, Xinying Xu
math.AP