On the regularity to the solutions of the Navier--Stokes equations via one velocity component

Abstract

We consider the regularity criteria for the incompressible Navier--Stokes equations connected with one velocity component. Based on the method from \cite{CaTi} we prove that the weak solution is regular, provided $ u_3 \in L^t(0,T; L^s(\R^3))$, $\frac 2t +\frac 3s \leq \frac 34 + \frac{1}{2s}$, $s> \frac {10}{3} $ or provided $ \nabla u_3 \in L^t(0,T; L^s(\R^3))$, $\frac 2t +\frac 3s \leq \frac{19}{12} + \frac 1{2s}$ if $s\in (\frac{30}{19},3]$ or $\frac 2t + \frac 3s \leq \frac 32 + \frac{3}{4s}$ if $s \in (3,\infty]$.

As a corollary, we also improve the regularity criteria expressed by the regularity of $\pder{p}{x_3}$ or $\pder{u_3}{x_3}$.

Keywords

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