On the one time-varying component regularity criteria for 3-D Navier-Stokes equations

In this paper, we consider the one time-varying component regularity criteria for local strong solution of 3-D Navier-Stokes equations. Precisely, if $\beta(t)$ is a piecewise $H^1$ unit vector from $[0,T]$ to $\Bbb{S}^2$ with finitely many jump discontinuities, we prove that if $\int_0^{T}\|u(t)\cdot \beta(t)\|_{\dot{H}^{\frac32}(\mathbb{R}^3)}^2\,dt<\infty,$ then the solution $u$ can be extended beyond the time $T.$ Compared with the previous results concerning one-component regularity criteria, here the unit vector $\beta(t)$ varies with time variable.