Well-Posedness and Optimal Time-Decay for Compressible MHD System in   Besov Space
  In this paper, firstly, we prove the global well-posedness of three dimensional compressible magnetohydrodynamics equations for some classes of large initial data, which may have large oscillation for the density and large energy for the velocity and magnetic field. Secondly, we prove the optimal time decay for the compressible magnetohydrodynamics equations with low regularity assumptions about the initial data. Especially, we can obtain the optimal $L^{2}$ time decay rate when the initial data small in the critical Besov space (no small condition in space $H^{N/2+1}$). When we calculate the optimal time decay rate, we use differential type energy estimates in homogeneous Besov space, evolution in negative Besov space and the well-posedness results proved in the first part. 
Junxiong Jia, Jigen Peng, Kexue Li
math.AP