  In this paper, we consider the regularity criterion for 3D incompressible
Navier-Stokes equations in terms of one directional derivative of the velocity
in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a
regular solution if the $\partial_3u$ satisfies $$\int^{T}_{0}
\frac{\left\|\left\|\left\|\partial_3 u(t) \right\|_{L^p_{x_1}}
\right\|_{L^q_{x_2}} \right\|^{\beta}_{L^{r}_{x_3}}} {1 +
\ln\left(\|\partial_3u \left(t\right)\|_{L^2} + e\right)}dt < \infty,$$ $\text
{ where } \frac{2}{\beta}+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 \text { and } 2
< p, q, r \leq \infty, 1-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right) \geq
0 $.

Analysis of PDEs