Navier-Stokes equations with regularity in one direction

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirao da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.