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

TL;DR

This paper investigates the local behavior of solutions to the Navier-Stokes equations near potential singular points, establishing concentration rates in weak Lebesgue spaces and introducing new regularity criteria.

Contribution

It introduces new concentration rate estimates for solutions near singularities and develops $ ext{ε}$-regularity criteria in $L^{p, ext{∞}}$ spaces, advancing understanding of singularity formation.

Findings

Established concentration rates for $L^{p, ext{∞}}$ norms near singular points.

Proved $ ext{ε}$-regularity criteria in $L^{p, ext{∞}}$ spaces.

Linked $L^{p, ext{∞}}$ spaces to Morrey type spaces for analysis.

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 $3\leq p\leq\infty$. 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.