On global regularity of 2D generalized magnetohydrodynamic equations

In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are −ν(−△)αu and −κ(−△)βb. We show that smooth solutions are global in the following three cases: α≥1/2,β≥1; 0≤α<1/2, 2α+β>2; α≥2, β=0. We also show that in the inviscid case ν=0, if β>1, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.