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.