Duality of Navier-Stokes to a one-dimensional system

TL;DR

The paper reframes Navier–Stokes turbulence as a linear quantum-like problem in loop space, introducing a momentum-loop equation whose exact, universal decaying-turbulence solution is the Euler ensemble. This ensemble admits a dual description as a string theory on a discrete target space of regular star polygons, with a random-walk dynamics over polygon vertices and an associated dual Wilson-loop amplitude; a No Explosion Theorem removes finite-time singularities under stochastic initial data. The work yields precise predictions for the decay of turbulence, vorticity statistics, and energy spectra, matching numerical and experimental data, and provides a rigorous link between fluid dynamics, number theory, and discrete string theories. It also clarifies how large Reynolds limits can differ from vanishing viscosity limits and discusses rich avenues for physical generalizations and mathematical directions. Overall, it offers a mathematically tractable, testable framework for turbulent NS dynamics with deep connections to geometry and quantum-like loop space methods.

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 $\vec{P}(θ, t)$, 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, \[ \left\langle \exp\left( \imath \oint dθ\, \vec{C}'(θ) \cdot \vec{v}(\vec{C}(θ, t)) \right) \right\rangle, \] reduces to a dual amplitude of this string theory with distributed external momentum proportional to $\vec{C}'(θ)/\sqrt{t}$.

Figures (8)

• Figure 1: Asymptotic trajectories of the time evolution for the loop functional inside the unit circle in the complex plane. The laminar flow is the yellow region on the circle close to $\Psi =1$. Three other flows are 1) hypothetical explosion, 2) decaying turbulence, and 3) special fixed point.
• Figure 2: The "hairpin" loop $C$ used in defining the pair correlation of vorticity. The little circles are the loop variations needed to bring down vorticity at two points in space. The backtracking contribution to the circulation cancels at vanishing separation between these parallel lines.
• Figure 3: The loop $C$ used to compute vorticity correlation functions. The endpoints of each spoke correspond to $\vec{C}(\theta_k) = \vec{r}_k$, while the center represents the center of mass, $\vec{r}_C = \frac{\sum_1^n \vec{r}_k}{n}$. Each spoke consists of two segments: one from the center to $\vec{r}_k$, and another returning to the center. Since the area enclosed by the loop is zero, the circulation vanishes, i.e., $\Gamma_C[v]=0$.
• Figure 4: regular star polygons for Euler ensembles of various $p,q$. The $\sigma_k$ variable indicates the direction of the random step of the link $k\leftrightarrow k+1$. The random walk could go several times around the polygon as long as it ends where it started.
• Figure 5: The world sheet of our discrete string made of regular star polygons with unit side. The red/green colors of the sides indicate random directions of random walk.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2