Fluid and Biodynamics Seminar: Duality of Navier-Stokes to a one-dimensional system

The Navier–Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear partial differential equations (PDEs) system. Despite their fundamental importance, their behavior in turbulent regimes remains incompletely understood, and their global regularity is still an open problem. Here, we reformulate the NS equations as a non-linear equation for the momentum loop P(ϴ,t), effectively reducing the original three-dimensional PDE to a one-dimensional problem. A key result of this reformulation is the derivation of a No Explosion Theorem, establishing that finite-time singularities do not occur under stochastic initial conditions. We also present an explicit analytical solution - the Euler ensemble - which describes the universal asymptotic state of decaying turbulence and is supported by numerical simulations and experimental validation.