Regularity criteria for NSE 5: $u_3,\om_3$

In [Zhang, Zujin. Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component. Czechoslovak Math. J. 68 (2018), no. 1, 219--225], we give an affirmative answer to an open problem in [Penel, Patrick; Pokorn\'y, Milan. Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math. 49 (2004), no. 5, 483--493], that is, whether or not we could obtain a regularity criterion involving only $u_3$ and $\om_3=\p_1u_2-\p_2u_1$. Our result reveals that if $$\bee\label{this} \bea u_3\in L^p(0,T;L^q(\bbR^3));&\quad \omega_3\in L^r(0,T;L^s(\bbR^3)),\\ \frac{2}{p}+\frac{3}{q}=1,\ 3<q\leq\infty;&\quad \frac{2}{r}+\frac{3}{s}=2,\quad \frac{3}{2}< s\leq \infty, \eea \eee$$ then the solution is smooth on $(0,T)$.