Regularity Criteria of The Incompressible Navier-Stokes Equations via Only One Entry of Velocity Gradient

Abstract

In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, $$ \partial _{3}u_{3}$$ . We show that if $$\partial _{3}u_{3}$$ satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor $$\partial _1 u_3$$ are also established, which covers some previous literature.