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 ∫ 0 T ‖ j ( . , t ) ‖ L 2 λ p λ p − 1 4 α λ p ( 2 α − 1 ) p + 2 d t < ∞ with 1 < α < 2 , 1 2 ≤ λ ≤ 1 , 2 ≤ p ≤ 2 λ , or the vorticity w ≜ ∇ × u satisfies ∫ 0 T ‖ w ( t , . ) ‖ BMO d t < ∞ with 1 < α < 2 .