Regularity criteria for the 3D MHD equations via one directional derivative of the pressure

In this paper, we consider the Cauchy problem for the 3D viscous MHD equations, and provide some regularity criteria involving only one directional derivative of the pressure, say ∂ 3 p . In particular, if ∂ 3 p ∈ L α ( 0 , T ; L β ( R 3 ) ) with  2 α + 3 β = 2 , 3 2 ≤ β ≤ 3 , then the solution remains smooth on [ 0 , T ] .