On the interior regularity criterion and the number of singular points to the Navier-Stokes equations

We establish some interior regularity criterions of suitable weak solutions for the 3-D Navier-Stokes equations, which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-\v{S}ver'{a}k’s criterion. We also show that if weak solution $u$ satisfies $$\mathrm{\parallel }u(\cdot ,t){\mathrm{\parallel }}_{L^p}\le C(-t{)}^{\frac{3-p}{2p}}$$ for some $3<p<\mathrm{\infty }$, then the number of singular points is finite.