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We study the global existence of classical solutions for two-dimensionalincompressible MHD system with only magnetic diffusion. By using thetime-weighted lower-order energy and uniformly bounded higher-order energyestimates, we prove the global existence result under the assumption that theinitial magnetic field is close enough to a constant magnetic vorticityequilibrium.

Global well-posedness for 2D inviscid and resistive MHD system near an equilibrium