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 $3\leq p\leq\infty$. 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_{t\to t_0^-}||u(t,x)-u(t)_{x_0,r}||_{L^{3,\infty}(B_r(x_0))}>\delta^*,\notag \end{align} and \begin{align} \limsup_{t\to t_0^-}(t_0-t)^{\frac{1}{\mu}}r^{\frac{2}{\nu}-\frac{3}{p}}||u(t)||_{L^{p,\infty}(B_r(x_0))}>\delta^*\notag for~3<p\leq\infty, ~\frac{1}{\mu}+\frac{1}{\nu}=\frac{1}{2}~and~2\leq\nu\leq\frac{2}{3}p,\notag \end{align}where $\delta^*$ is a positive constant independent of $p$ and $\nu$. 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.