Duality of Navier-Stokes to a One-Dimensional System

The Navier-Stokes (NS) equations describe fluid motion via a high-dimensional, nonlinear system of partial differential equations (PDEs). Despite their fundamental role, their behavior in turbulent regimes remains poorly understood, and their global regularity is an open Millennium Problem. Here, we show that the NS system can be transformed to a nonlinear equation for the momentum loop P→(θ,t), reducing the original problem in R3 to a one-dimensional problem. A key consequence is the No Explosion Theorem, which rules out finite-time singularities for stochastic initial conditions, suggesting a pathway to proving NS regularity for deterministic data. Additionally, we derive an exact solution of the momentum loop equation, the Euler ensemble, which describes the asymptotic state of decaying turbulence. This solution, validated by numerical and experimental data, replaces heuristic turbulence models with an analytically derived structure. While the Euler ensemble solves a specific boundary problem, its universality as an attractor remains an open question. By reducing NS to a nonlinear equation on S1, this work opens new pathways toward resolving turbulence and fluid mechanics’ fundamental challenges.