The localized characterization for the singularity formation in the Navier-Stokes equations

This paper is concerned with the localized behaviors of the solution $u$ to the Navier-Stokes equations near the potential singular points. We establish the concentration rate for the $L^{p,infty}$ norm of $u$ with $3leq pleqinfty$. Namely, we show that if $z_0=(t_0,x_0)$ is a singular point, then for any $r>0$, it holds begin{align} limsup_{tto t_0^-}||u(t,x)-u(t)_{x_0,r}||_{L^{3,infty}(B_r(x_0))}>delta^*, otag end{align} and begin{align} limsup_{tto t_0^-}(t_0-t)^{frac{1}{mu}}r^{frac{2}{ u}-frac{3}{p}}||u(t)||_{L^{p,infty}(B_r(x_0))}>delta^* otag for~3<pleqinfty, ~frac{1}{mu}+frac{1}{ u}=frac{1}{2}~and~2leq uleqfrac{2}{3}p, otag end{align}where $delta^*$ is a positive constant independent of $p$ and $ u$. Our main tools are some $varepsilon$-regularity criteria in $L^{p,infty}$ spaces and an embedding theorem from $L^{p,infty}$ space into a Morrey type space. These are of independent interests.