In this talk, we consider a two-dimensional non-resistivity MHD system describing the evolution of viscous compressible and electrically conducting fluids under the action of a vertical magnetic field, with non-monotone pressure law and density-depending viscosity λ =λ(ρ). Using an approximate scheme and the compactness method which Bresch and Jabin proposed in (Bresch and Jabin, 2018), we prove the global existence of weak solutions. (Based on the work with Yu Liu).