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 $$eelabel{u_3,p_3u_1,p_3u_2} u_3in L^infty(0,T;L^3(bR^3))mbox{ and }p_3bu_hin L^e(0,T;L^al(bR^3)),quad f{2}{e}+f{3}{al}=2,quad 2leq alleq infty, eee$$ could ensure the smoothness of the solution. This improves the result $$eelabel{Qian16} u_3in L^infty(0,T;L^3(bR^3))mbox{ and }p_3bu_hin L^e(0,T;L^al(bR^3)),quad f{2}{e}+f{3}{al}=2,quad 2leq alleq 3, eee$$ 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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