  Whether or not classical solutions of the 2D incompressible MHD equations
without full dissipation and magnetic diffusion can develop finite-time
singularities is a difficult issue. A major result of this paper establishes
the global regularity of classical solutions for the MHD equations with mixed
partial dissipation and magnetic diffusion. In addition, the global existence,
conditional regularity and uniqueness of a weak solution is obtained for the 2D
MHD equations with only magnetic diffusion.
