A logarithmically improved regularity criterion for the MHD equations in terms of one directional derivative of the pressure

In this paper, a logarithmically improved regularity criterion for the incompressible magnetohydrodynamics equation is established in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure (Formula presented.) satisfies the logarithmical Serrin-type condition (Formula presented.) then the solution (u, b) remains smooth on (Formula presented.). Compared to the Navier–Stokes result, there is a logarithmic correction involving b in the denominator. This is an extension of earlier regularity results in the Serrin’s type space (Formula presented.) with 3/2 < λ ≤ ∞.