The regularity criterion for 3D Navier-Stokes Equations

In this article, we establish sufficient conditions for the regularity of solutions of Navier-Stokes equations based on one of the nine entries of the gradient tensor. We improve the recently results of C.S. Cao, E.S. Titi (Arch. Rational Mech.Anal. 202 (2011) 919-932) and Y. Zhou, M. Pokorn$\acute{y}$ (Nonlinearity 23, 1097-1107 (2010)).