On the possible time singularities for the 3D Navier-Stokes equations

We prove a local-in-time regularity criterion for the 3D Navier-Stokes equations. In particular, it follows from the criterion that the Hausdorff dimension of possible singular times of Leray-Hopf weak solutions $u\in L^r_t B^\alpha_{s,\infty}$forsome$\alpha>0$,$ s > 3$and$r> 2 $islessthan$\frac{r}{2}(\frac{3}{s} + \frac{2}{r} -\alpha-1 )$. The main contribution is that we do not assume the suitability of weak solutions.