Regularity criterion for weak solutions to the Navier-Stokes involving one velocity and one vorticity components

In this note, we are devoted to study the conditional regularity for the three dimensional Navier-Stokes in terms of the Morrey and $BMO$ spaces. More precisely, we show that if $u$ is a weak solution and $u_{3}\in L^{2}(0,T;BMO(\mathbb{R}^{3}))$ and $\omega _{3}\in L^{ \frac{2}{2-r}}(0,T;\mathcal{\dot{M}}_{2,\frac{3}{r}}(\mathbb{R}^{3}))$ with $0<r<1$, then $u$ is regular on $(0,T]$. This improves the available result by Zhang (2018) with $u_{3}\in L^{2}(0,T;L^{\infty }(\mathbb{R}^{3}))$ and $\omega _{3}\in L^{\frac{2}{2-r}}(0,T;L^{\frac{3}{r}}(\mathbb{R}^{3}))$ with $0<r<1$.