The regularity criterion for 3D Navier-Stokes equations involving one velocity gradient component

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 recent results of C.S. Cao, E.S. Titi [C.S. Cao, E.S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal. 202 (2011) 919-932] and Y. Zhou, M. Pokorny [Y. Zhou, M. Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity 23 (2010) 1097-1107]. (c) 2012 Elsevier Ltd. All rights reserved.