In [Zhang, Zujin. An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field. Bull. Math. Sci. 8 (2018), no. 1, 33--47] we have improved the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously. The result reads: the condition
$$\bee\label{me}\p_3\bbu\in L^p(0,T;L^q(\bbR^3)),\quad \frac{2}{p}+\frac{3}{q}=2,\quad \frac{3\sqrt{37}}{4}-3\leq q\leq 3\eee$$
could ensure the regularity of the solution.

see https://link.springer.com/article/10.1007/s13373-016-0098-x.