On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component

Abstract

We improve the regularity criterion for the incompressible Navier--Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results by Cao and Titi and Kukavica and Ziane In particular, for $s\in [2,3]$ we get that the solution is regular if $\nabla u_3 \in L^t(0,T; L^s(\R^3))$, $\frac 2t + \frac 3s \leq \frac{23}{12}$.

The criteria for $s$ outside this interval are weaker.

Keywords

• regularity of solutions
• Navier--Stokes equations
• one velocity component