Regularity criteria for the Navier-Stokes equations based on one or two items of the velocity gradient

We study the regularity criteria for the incompressible Navier-Stokes equations based on either one item of the velocity gradient, ∂1u3 or ∂3u3 or on two items of the velocity gradient, ∂2u3, ∂3u3, or ∂1u3, ∂2u3. We improve and/or extend several results from the literature by the use of two versions of the generalized Troisi inequality.