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

Abstract

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 $3\le p\le \infty$. 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))}>δ^*,\notag \end{align} and \begin{align} \limsup_{t\to t_0^-}(t_0-t)^{\frac{1}μ}r^{\frac{2}ν-\frac{3}{p}}||u(t)||_{L^{p,\infty}(B_r(x_0))}>δ^*\notag for~3<p\leq\infty, ~\frac{1}μ+\frac{1}ν=\frac{1}{2}~and~2\leqν\leq\frac{2}{3}p,\notag \end{align}where $\hat{I}{´}^{\ast }$ is a positive constant independent of $p$ and $\hat{I}½$. Our main tools are some $\epsilon$-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.

arXiv admin note: text overlap with arXiv:2107.04157