  Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D
Navier-Stokes equations are obtained. The results show that a weak solution $u$
becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $
\nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T;
\dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where
$\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient
operator. Besides, we also consider the anisotropic regularity criterion for
the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we
also get a further regularity criterion, when give the sufficient condition on
$\partial_3u_3$.
