arXiv 2107.045972021-07-09math.AP

Navier-斯托克斯方程奇异性形成的局部化特征

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

W. Tan

备注：: arXiv admin note: text overlap with arXiv:2107.04157

摘要：

本文研究了Navier-斯托克斯方程解$u$在位势奇点附近的局部化行为。我们用$3\leq p\leq\infty$建立了$u$的$L^{p,\infty}$标准的集中率。也就是说，我们表明，如果$z_0=(t_0,x_0)$是一个奇点，那么对于任何$r>0$，它持有\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}和\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}，其中$δ^*$是一个独立于$p$和$ν$的正常数。我们的主要工具是$L^{p,\infty}$空间中的一些$\varepsilon$ -正则性准则和从$L^{p,\infty}$空间到Morrey型空间的嵌入定理。这些都是独立的利益。

英文摘要：

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))}>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 $δ^*$ is a positive constant independent of $p$ and $ν$. 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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