Incompressible limit of global solutions to the compressible magnetohydrodynamics equations for large initial data in critical Besov spaces

We study global existence of solutions to slightly compressible magnetohydrodynamics equations. If the magnetic field and the incompressible part of the velocity generate a global solution to the incompressible magnetohydrodynamics equations, we show that compressible magnetohydrodynamics equations admit a global regular solution for arbitrary large initial date in critical Besov spaces provided that the Mach number is sufficiently small. Furthermore, the convergence rate, with respect to Mach number, of solutions to the original equations toward that of the limit equations is established. The key issue is the construction of a Lp type energy functional Mp,qε,α which captures characters of different frequency and can be arbitrary small for suitable parameters.