In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{ a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely $\partial_3 \mathbf{ u} \in L^p((0,T); L^q(\mathbb{ R}^3)), ~\frac{ 2}{ p} + \frac{ 3}{ q} = 2, ~\frac{ 3}{ 2}<q\leq 6$. Moreover, we extend the range of $q$ while the solution is axisymmetric, i.e. the axisymmetric solution $\mathbf{ m}{ u}$ is regular in $(0,T]$, if $ \partial_3 u^3 \in L^p((0,T); L^q(\mathbb{ R}^3)), ~\frac{ 2}{ p} + \frac{ 3}{ q} = 2, ~\frac{ 3}{ 2}<q< \infty$.
Categories: math.AP