轴对称Navier-Stokes方程具有Euler长度的正则性

中文摘要 AI 辅助

我们证明了三维Navier-Stokes方程的轴对称适定弱解的局部正则性，这些解在终止时间$t=0$之前是光滑的，并在一个消失的长度尺度$\ell(t)$上满足Type~II逐点界。我们称$\ell(t)$为Euler长度，如果它是非递增的，满足一个倍增条件，并且当$t\to0^-$时，$\ell(t)\to0$且$(-t)/\ell(t)^2\to0$。这包括幂律$\ell(t)=(-t)^\gamma$（其中$0<\gamma<1/2$）以及对抛物型长度的对数放大。我们的主要结果表明，对于所有$t\in (-1,0)$，形如$|u(\cdot,t)| \leq C\ell(t)/(-t)$和$|\nabla^2 u(\cdot,t)|\leq C/((-t)\ell(t))$的局部界蕴含正则性。证明将我们先前论文~\cite{CIV26}中的环量和位势涡度论证改编应用于通过放大从重标度涡度系统获得的古老极限。我们证明了二阶导数的先验假设可以被方位角涡度的Hölder界及其位势涡度的相应界所替代。

英文摘要

We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time $t=0$, and satisfy Type~II pointwise bounds at a vanishing length scale $\ell(t)$. We say $\ell(t)$ is an Euler length if it is non-increasing, satisfies a doubling condition, and if $\ell(t)\to0$ and $(-t)/\ell(t)^2\to0$ as $t\to0^-$. This includes power laws $\ell(t)=(-t)^γ$ with $0<γ<1/2$, and logarithmic enlargements of the parabolic length. Our main result shows that local bounds of the type $|u(\cdot,t)| \leq C\ell(t)/(-t)$ and $|\nabla^2 u(\cdot,t)|\leq C/((-t)\ell(t))$ for all $t\in (-1,0)$ imply regularity. The proof adapts the circulation and potential-vorticity argument of our earlier paper~\cite{CIV26} to ancient limits obtained from the rescaled vorticity system by zooming in. We show that the second derivative a priori assumption may be replaced by a Hölder bound on the azimuthal vorticity and a corresponding bound on its potential vorticity.

发表机构

• Princeton University（普林斯顿大学）
• Temple University（天普大学）
• New York University（纽约大学）

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