具有解析外力项的三维Navier-Stokes方程渐近轴对称解的正则性

中文摘要 AI 辅助

OpenAI~\cite{OpenAIManuscript} 最近宣布证明了在 $C^\infty$ 光滑体积力存在的情况下，三维Navier-Stokes方程有限时间奇点形成的证明。~\cite{OpenAIManuscript} 中的构造具有几个关键特征，我们特别指出：(i) 解的角向平均值满足特定的各向异性的Type II界；(ii) 在坍缩的核心区域内，解是严格轴对称的。在本文中，我们考虑在实解析体积力存在的情况下三维Navier-Stokes方程的解，并假设这些解满足上述性质 (i) 和 (ii)。我们证明这样的解在假定的奇点处实际上是正则的。因此，在~\cite{OpenAIManuscript} 的构造中，以及在任何具有性质 (i) 和 (ii) 且其力在奇点时刻之前在 $C^2$ 中保持有界的构造中，该力既不能在奇点附近恒为零，也不能在空间变量中是实解析的，且不能在时间上局部一致。证明的主要思想受到我们早期工作~\cite{CIV26} 和姊妹篇~\cite{CIVEulerLength} 的启发：利用先验界提供的各向异性长度尺度放大假定的奇点，我们得到古代极限，其PDE演化施加了额外的刚性。

英文摘要

OpenAI~\cite{OpenAIManuscript} has recently announced a proof of finite time singularity formation for the 3D Navier-Stokes equations, in the presence of a $C^\infty$-smooth body force. The construction in~\cite{OpenAIManuscript} has a few key features, among which we single out: (i) the angular mean of the solution satisfies specific Type II bounds which are anisotropic; (ii) the solution is exactly axisymmetric in a collapsing core region. In this paper we consider solutions of the 3D Navier-Stokes equations in the presence of a real-analytic body force, and assume that these solutions satisfy properties (i) and (ii) above. We prove that such solutions are in fact regular at the putative singular point. As a consequence, in the construction of~\cite{OpenAIManuscript}, and in any construction with properties (i) and (ii) whose force remains bounded in $C^2$ up to the singular time, the force can neither vanish identically near the singular point, nor be real analytic in the space variables, locally uniformly in time. The main idea of the proof is inspired by our earlier work~\cite{CIV26} and the companion paper~\cite{CIVEulerLength}: zooming-in at the putative singularity using the anisotropic length scales provided by the a priori bounds, we arrive at ancient limits whose PDE evolution imposes additional rigidity.

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）

机构由 AI 辅助整理，请以论文原文为准。