发表机构
UAB（阿拉巴马大学伯明翰分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三维临界Navier-Stokes方程中涡量方向的正则性，通过分析几何PDE，证明在核心处方向的对数振荡衰减由内尺度时间模量和切向应变控制，且应变仅在方向非特征向量时起作用，与经典准则形成对比。
AI 中文摘要
我们分离并分析了三维不可压缩Navier-Stokes方程（NSE）中控制涡量方向演化的几何偏微分方程（PDE），并将其限制在临界空间点奇异性的情形下，此时涡量大小以$O(|x|^{-2})$的形式集中，处于临界Lorentz空间$L^{3/2, \infty}$中。该PDE由球面上的调和映射热流（HMHF）补充流体输运、交叉扩散和切向应变组成。问题在于NSE机制能否传播方向局部平均振荡的对数衰减——即条件$\xiVec \in \bmo_{1/|\log r|}$（在此设定下，配套论文中已证明该条件可阻止有限时间爆破）。关键观察是：从粘性交叉扩散中分解出的$O(|x|^{-2})$浓度在核心处产生向外径向漂移$4\nu\\, x/|x|^2$，且HMHF非线性对于量$\frac12|\xiVec - e|^2$是无害的，该量在任何半球上都是子解。由此得出传递定理：核心处方向在$\bmo_{1/|\log r|}$意义下的正则性，在直至奇异时间的一致意义上，由方向在内尺度上的时间对数模量以及物理应变共同控制。关键在于，应变仅通过其切向分量$P_{\xiVec^\perp} S\xiVec$进入方向方程，而当方向恰好是应变张量的特征向量时该分量精确为零。由于这包括携带最大拉伸的特征向量，该结果与经典的几何正则性准则形成对比，后者建立在耗尽完整涡拉伸项的基础上，因此仅限于弱拉伸的构型。
英文摘要
We isolate and analyze the geometric PDE governing the evolution of the vorticity direction in the 3D incompressible (unforced) Navier-Stokes equations (NSE), restricted to the case of a critical spatial point singularity where the vorticity magnitude concentrates as $O(|x|^{-2})$, inhabiting the critical Lorentz space $L^{3/2, \infty}$. The PDE consists of the Harmonic Map Heat Flow (HMHF) into the sphere supplemented with the fluid transport, cross-diffusion and tangential strain. The question is whether the NSE mechanics can propagate logarithmic decay of the local mean oscillations of the direction -- the condition $\xiVec \in \bmo_{1/|\log r|}$ which (in this setting) was shown in the companion paper to prevent finite time blow-up. The key observations are that the $O(|x|^{-2})$ concentration, factored out of the viscous cross-diffusion, generates an outward radial drift $4ν\, x/|x|^2$ at the core and that the HMHF nonlinearity is harmless for the quantity $\frac12|\xiVec - e|^2$, which is a subsolution on any hemisphere. This yields a transfer theorem: the $\bmo_{1/|\log r|}$ regularity of the direction at the core, uniformly up to the singular time, is controlled by a logarithmic modulus in time of the direction at the inner scale, together with the physical strain. Crucially, the strain enters the direction equation only through its tangential component $P_{\xiVec^\perp} S\xiVec$, which vanishes precisely when the direction is an eigenvector of the strain tensor. Since this includes the eigenvector carrying the maximal stretching, the result is in contrast to the classical geometric regularity criteria which are built on depleting the full vortex-stretching term and thus confined to configurations of weak stretching.
Commentscross-referencing a companion paper arXiv:2607.08866 plus a couple of clarifications
