Regularity criterion for 3D Navier-Stokes Equations in Besov spaces

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 ${\mathrm{\nabla }}_{h}u$ (or $\mathrm{\nabla }{u}_3$) 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 ${\mathrm{\partial }}_3{u}_3$.