We follow the approach of  to study the N-dimensional generalized MHD system with fractional Laplacians as dissipative and diffusive terms in various anisotropic spaces. In particular, we obtain small initial data results with anisotropic Sobolev space type norms for which, depending on the power of the fractional Laplacians, we may decrease the regularity index in many directions to zero or even negative, in the expense of increasing the rest. Similar results in anisotropic Besov type spaces are also obtained.
"On the global well-posedness of N-dimensional generalized MHD system in anisotropic spaces." Adv. Differential Equations 19 (3/4) 201 - 224, March/April 2014.