Regularity criteria for NSE 6: $u_3,\p_3u_1,\p_3u_2$

In [Zujin Zhang, Jinlu Li, Zheng-an Yao, A remark on the global regularity criterion for the 3D Navier-Stokes equations based on end-point Prodi-Serrin conditions, Applied Mathematics Letters, 83 (2018), 182—187], we take full advantage of the regularity of the vertical velocity component, and show that $$\bee\label{u_3,p_3u_1,p_3u_2} u_3\in L^\infty(0,T;L^3(\bbR^3))\mbox{ and }\p_3\bbu_h\in L^\be(0,T;L^\al(\bbR^3)),\quad \f{2}{\be}+\f{3}{\al}=2,\quad 2\leq \al\leq \infty, \eee$$ could ensure the smoothness of the solution. This improves the result $$\bee\label{Qian16} u_3\in L^\infty(0,T;L^3(\bbR^3))\mbox{ and }\p_3\bbu_h\in L^\be(0,T;L^\al(\bbR^3)),\quad \f{2}{\be}+\f{3}{\al}=2,\quad 2\leq \al\leq 3, \eee$$ in [Qian, Chenyin. A remark on the global regularity for the 3D Navier-Stokes equations. Appl. Math. Lett. 57 (2016), 126--131] significantly.

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