Regularity criteria for the generalized viscous MHD equations

In this paper, we consider regularity criteria for solutions to the 3D generalized MHD equations with fractional dissipative term -(-) α u for the velocity field and -(-) β b for the magnetic field.For the case α = β, it is proved that if the velocity field belongs to L p,q with 2α/p + 3/q 2α -1 or the gradient of velocity field belongs to L p,q with 2α/p + 3/q 3α -1 on [0, T ], then the solution remains smooth on [0, T ].The significance is that there is no restriction on the magnetic field.Moreover, the norms u L p,q and Λ α u L p,q are scaling dimension zero for 2α/p + 3/q = 2α -1 and 2α/p + 3/q = 3α -1 respectively.For 1 β α, we find that the minimum sum of α and β to guarantee the global existence of smooth solutions is 5/2.Furthermore, we show that the weak solution actually is strong if the corresponding vorticity field ω = ∇ × u satisfies certain condition in the high vorticity region.