The Evolutionary Layer-by-Layer Gauge Method: A Geometric Construction of Global Regularity for the Navier–Stokes Equations

This paper presents a complete geometric construction of global regular solutions for the three-dimensional Navier–Stokes equations. Building on the published classification of degenerate solutions [1] and the continuous gauge method developed in [2], we introduce a constructive, layer-by-layer approach that resolves the fundamental issue of non-commutation of the Laplacian with diffeomorphisms.The method combines two key ideas:1. A geometric decomposition of arbitrary 3D flows into simpler components: first eliminating one spatial coordinate to obtain a 2D flow, then transformingthe 2D flow into an axisymmetric flow without swirl. Both steps are performed using time-dependent diffeomorphisms, ensuring consistency with the viscous term.2. An analytical framework based on a modified entropy functional that controls both the velocity field and the gauge deformation. This framework, developed in [2], provides the a priori estimates needed to prevent finite-time blow-up.The classification theorem [1] plays a crucial role: it shows that the only geometrically rigid degenerate case (rank 1, where streamlines are straight lines) cannot arise in solutions that are gauge-equivalent to a non-trivial axisymmetric flow. This eliminates the only potential obstruction to regularity. Using Fourier expansion in the angular variable, we prove that the class of fields admitting such an evolutionary gauge construction is dense. Uniform entropy estimates then show that this class is also closed, hence it coincides with the space of all smooth divergence-free fields. Consequently, every smooth initial data yields a globally regular solution.This work resolves the regularity problem in the affirmative, providing a constructive geometric proof that avoids the analytical difficulties that have hindered previous approaches.