Duality of Navier-Stokes to a one-dimensional system

Abstract

The Navier--Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear system of partial differential equations (PDEs). 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 nonlinear equation for the momentum loop $\vec{P}(\theta, t)$, effectively reducing the original three-dimensional PDE to a one-dimensional problem. We 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. This Euler ensemble is equivalent to a string theory with discrete target space given by a set of regular star polygons, with additional Ising (Fermi) degrees of freedom at the vertices. This string theory can also be interpreted as a random walk on regular star polygons. The Wilson loop for turbulence, \[ \left\langle \exp\left( \imath \oint d\theta\, \vec{C}'(\theta) \cdot \vec{v}(\vec{C}(\theta, t)) \right) \right\rangle, \] reduces to a dual amplitude of this string theory with distributed external momentum proportional to $\vec{C}'(\theta)/\sqrt{t}$.