Global Regularity of the Three-Dimensional Navier–Stokes Equations via Geometric Regularization of the Universal Transport Equation

This paper addresses the existence and uniqueness of global smooth solutions to the three-dimensional incompressible Navier–Stokes equations. By constructing the Universal Transport Equation (UTE), the original system is lifted to a complete Riemannian manifold of uniformly negative sectional curvature. This lifting introduces a second-order spectral dis sipation mechanism induced by the curvature tensor, thereby controlling the high-frequency energy accumulation caused by vortex stretching nonlinearity. In Fourier spectral space, a differential inequality for the vorticity energy spectrum is established, and its uniform boundedness is proved via Bernoulli-type comparison principles. Combined with the Beale– Kato–Majda blow-up criterion, finite-time singularities are excluded, yielding the existence and uniqueness of global smooth solutions. A core theorem statement in the Lean 4 formal verification framework is also provided. 本文研究三维不可压缩 Navier–Stokes 方程全局光滑解的存在唯一性问题。通过构造通用 输运方程(Universal Transport Equation, UTE),将原系统提升至具有一致负截面曲率的完备 黎曼流形。该提升引入由曲率张量诱导的二阶谱耗散机制,从而控制涡拉伸非线性引发的高 频能量累积。在傅里叶谱空间内,建立涡量能谱的微分不等式,并借助 Bernoulli 型比较原理 证明其一致有界性。结合 Beale–Kato–Majda 爆破准则,排除有限时间奇点,进而获得全局光 滑解的存在性与唯一性。本文同时给出 Lean 4 形式化验证框架的核心定理声明。