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

Abstract

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 ${\mathbb{S}}^2$ with finitely many jump discontinuities, we prove that if ${\int }_0^T\Vert u(t)\cdot \beta (t){\Vert }_{{\dot{H}}^{\frac{3}{2}}({\mathbb{R}}^3)}^{2}dt<\infty ,$ 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.