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
研究三维不可压缩纳维-斯托克斯方程临界点奇点，提出几何分析机制，通过对数加权空间及相关估计等，使涡度拉伸耗散，将对数增益转移到速度场，避免有限时间爆破。
AI 中文摘要
我们提出了一种几何分析机制，用于抑制三维不可压缩纳维-斯托克斯方程中临界点奇点的有限时间奇点，这些奇点表现出涡度的\(L^{3/2, \infty}\)空间集中。我们证明，如果涡度方向局部位于有界平均振荡的对数加权空间\(\mathrm{bmo}_{1/|\log r|}\)中（该空间不满足迪尼条件，因此允许剧烈振荡缺陷），非线性涡旋拉伸将从根本上被耗尽。通过分离精确的单向几何抵消，我们将拉伸特征值重铸为奇异积分换位子。利用局部化的科伊夫曼-罗奇伯格-韦斯估计与二进BMO尾部界，我们证明拉伸势在收缩的超水平集上作为对数包络消失。这种耗散通过内插德乔治能量方法将涡度大小强制进入亚临界洛伦兹-齐格蒙德空间。随后，对数增益被转移到速度场，迫使局部一维稀疏的几何尺度低于空间解析性的均匀半径，最终通过调和测度最大原理避免有限时间爆破。
英文摘要
We present a geometric-analytic mechanism for the suppression of finite-time singularities in the 3D incompressible (unforced) Navier-Stokes equations for critical point singularities exhibiting $L^{3/2, \infty}$ spatial concentration of vorticity. We demonstrate that if the vorticity direction resides locally in a logarithmically weighted space of bounded mean oscillations, $\mathrm{bmo}_{1/|\log r|}$ -- a space failing the Dini condition and thus permitting wild oscillatory defects -- the non-linear vortex stretching is fundamentally depleted. By isolating a unidirectional geometric cancellation, we recast the stretching eigenvalue as a singular integral commutator. Utilizing a localized Coifman-Rochberg-Weiss estimate coupled with dyadic BMO tail bounds, we prove the stretching potential vanishes as a logarithmic envelope on shrinking super-level sets. This depletion forces the vorticity magnitude into a sub-critical Lorentz-Zygmund space via interpolated De Giorgi energy method. The logarithmic gain is subsequently transferred to the velocity field, forcing the geometric scale of local 1D sparseness below the uniform radius of spatial analyticity, ultimately averting the finite-time blow-up via the harmonic measure maximum principle.
Commentscross-referencing a companion paper arXiv:2609.05720 plus a couple of clarifications
