We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008)] and Kukavica and Ziane [see “Navier-Stokes equations with regularity in one direction,” J. Math. Phys. 48, 065203 (2007)]. In particular, for $s∊\left[2,3\right]$ , we get that the solution is regular if $\nabla u_3∊L^t\left(0,T;L^s\left({\mathbb{R}}^3\right)\right)$ , $2/t+3/s\le \frac{23}{12}$ .

: In this paper, various logarithmically improved regularity criteria for the 3D MHD equations are established in terms of the velocity field and pressure respectively. These regularity criteria can be regarded as log in time improvements of the standard Serrin's criteria established before.