Inverse Cascade Theorems for 2D, Axisymmetric, and Locally Quasi-Two-Dimensional Navier–Stokes Flows

We present a mathematical theory of the inverse energy cascade in incompressible Navier--Stokes flows. The central result is a rigorous proof that the inverse Kraichnan cascade---the transfer of kinetic energy from small to large scales---operates in three distinct regimes: purely two-dimensional flows, three-dimensional axisymmetric flows without swirl, and locally quasi-two-dimensional regions of arbitrary three-dimensional flows. The theory provides the missing mathematical foundation for the cascade dynamics observed in numerical simulations, laboratory experiments, and geophysical flows. The proof proceeds in five principal stages. First, we introduce the gauge enstrophy, a modification of the classical enstrophy weighted by a Lagrangian radial coordinate tied to the geometry of stream function level sets. We prove that the gauge enstrophy is an exact invariant of the two-dimensional Euler equations and that its dissipation in the Navier--Stokes equations is concentrated at the smallest scales, vanishing on large scales in the limit of zero viscosity. This provides the approximate conservation law necessary to extend the Fj{\o}rtoft argument to viscous flows. Second, we prove the inverse cascade theorem for 2D Navier--Stokes flows, establishing that the spectral energy flux is negative throughout the inertial range and that the energy spectrum follows the Kraichnan--Kolmogorov scaling. Third, we prove the isomorphism between axisymmetric three-dimensional flows without swirl and two-dimensional flows, extending the inverse cascade to this class of 3D flows. Fourth, we extend the analysis to arbitrary three-dimensional flows using the layer-by-layer gauge method, proving that the inverse cascade operates locally in any region where the velocity gradient has rank two. We establish the global vortex stratification theorem, which decomposes the quasi-two-dimensional subdomain into disjoint Morse--Smale cells separated by a singular set of measure zero, each supporting an independent inverse cascade. Fifth, we identify the critical threshold for the breakdown of the inverse cascade as the KAM winding index reaching the value twenty-eight, derived from the Lorenz attractor dynamics. Below this threshold, KAM tori protect the two-dimensional coherent dynamics; at the threshold, the strange attractor becomes globally attracting, three-dimensional perturbations are amplified, and the direct Kolmogorov cascade is triggered.