The Continuous Gauge Method: A Geometric Framework for the Navier–Stokes Regularity Problem

This paper introduces the Continuous Gauge Method, a geometric framework for studying the regularity of solutions to the three-dimensional Navier–Stokes equations. The method is based on deforming arbitrary initial data to axisymmetric initial data without swirl through a one-parameter family of diffeomorphisms, while simultaneously constructing a gauge field that tracks this deformation over time.In fact, this method is a modification of the method for investigating solutions for Ricci flows, developed and applied by Dennis DeTurck (see [1]) and subsequently called ”DeTurck’s trick”.The main results are:1. A geometric incompatibility theorem showing that rank-1 degenerate solutions cannot be obtained from non-trivial axisymmetric flows without swirl via smooth diffeomorphisms (Theorem 3.3).2. A Nash–Moser implicit function theorem argument establishing that a neighborhood of any axisymmetric field consists of gaugeable fields (Theorem 4.2).3. An entropy functional that provides analytical control over both the velocity field and the gauge deformation, together with a complete derivation of itsevolution inequality (Appendix B).4. A compactness theorem for blow-up sequences that preserves the gauge structure in the limit (Theorem 6.2).5. A proof that for any gaugeable initial data, the corresponding Navier–Stokes solution is globally regular (Theorem 6.6).6. A density argument showing that the gaugeable class is dense in the space of all smooth divergence-free fields (Theorem 7.6).7. A closure argument showing that the gaugeable class is closed (Section 8), which together with density implies that every smooth divergence-free field is gaugeable.Combining these results yields a complete proof that all smooth, divergence-free initial data for the three-dimensional Navier–Stokes equations give rise to unique, smooth solutions that exist for all time.本文介绍了连续规范方法（Continuous Gauge Method），这是用于研究三维纳维-斯托克斯方程（Navier–Stokes equations）解的正则性的几何框架。该方法通过单参数微分同胚（diffeomorphisms）族，将任意初始数据变形为无旋轴对称初始数据，同时构建规范场（gauge field）以追踪该变形随时间的演化过程。事实上，该方法是对丹尼斯·德特克（Dennis DeTurck）提出并应用的里奇流（Ricci flows）解研究方法的改进（参见文献[1]），该方法后被称为“德特克技巧（DeTurck’s trick）”。 本文的主要研究结果如下： 1. 几何不相容性定理：证明了通过光滑微分同胚，无法从非平凡无旋轴对称流得到秩1退化解（定理3.3）。 2. 基于纳什-莫泽隐函数定理（Nash–Moser implicit function theorem）的论证：确立了任意轴对称场的邻域均属于可规范场范畴（定理4.2）。 3. 提出可同时对速度场与规范变形进行解析控制的熵泛函（entropy functional），并完整推导了其演化不等式（附录B）。 4. 针对爆破序列（blow-up sequences）的紧性定理：该定理在极限情况下保留规范结构（定理6.2）。 5. 证明对于任意可规范初始数据，对应的纳维-斯托克斯方程解均具有全局正则性（定理6.6）。 6. 稠密性论证：证明可规范场类在全体光滑无散场（divergence-free fields）空间中稠密（定理7.6）。 7. 闭包论证：证明可规范场类是闭集（第8节），结合前述稠密性结论可推导出全体光滑无散场均为可规范场。 综合上述结果，本文完整证明了三维纳维-斯托克斯方程的所有光滑、无散初始数据，均可生成唯一且全局存在的光滑解。