Two regularity criteria to the 2D generalized MHD equations with zero magnetic diffusivity

In this paper, two regularity criteria to the two-dimensional incompressible generalized magnetohydrodynamics equations with zero magnetic diffusivity are given. It is proved that the solution remains smooth on [0,T] provided that the current j≜∇×b satisfies∫0T‖j(.,t)‖L2λpλp−14αλp(2α−1)p+2dt<∞ with 1<α<2, 12≤λ≤1, 2≤p≤2λ, or the vorticity w≜∇×u satisfies∫0T‖w(t,.)‖BMOdt<∞ with 1<α<2.