  The existence of global-in-time classical solutions to the Cauchy problem of
compressible magnetohydrodynamic flows with zero magnetic diffusivity is
considered in two dimensions. The linear structure is a degenerated
hyperbolic-parabolic system. The solution is constructed as a small
perturbation of a constant background in critical spaces. The deformation
gradient is introduced to decouple the subtle coupling between the flow and the
magnetic field. The $L^1$ dissipation for the velocity is obtained, and the
$L^2$ dissipations for the density and the magnetic field are also achieved.
