This work has demonstrated a duality between classical incompressible fluid mechanics in Euclidean space ℝ₃ and nonlinear dynamics in loop space. By reformulating the Navier-Stokes equations in terms of loop dynamics, we uncover universal properties of turbulent flows, providing new insights into the structure of decaying turbulence. Key contributions of this study include:
- The classical Navier-Stokes equations, when supplemented by thermal fluctuations, can be reformulated as a 1 + 1 dimensional equation for the momentum loop trajectory F(θ, t). The loop functional is related to the momentum loop via an explicit formula, providing a direct connection between turbulence and loop dynamics.
- The viscosity vanishes from the momentum loop equation, rendering the momentum loop trajectories completely universal. In this framework, the Reynolds number becomes a property of the initial data F(θ, 0), corresponding to a stochastic loop in ℝ₃.
- A degenerate fixed point, F(θ, t) = F_⋆(θ), is associated with the decaying turbulence solution. This fixed point corresponds to a periodic random walk on a regular star polygon with infinitely many steps and is analytically characterized by the Euler ensemble.
- The Euler ensemble is equivalent to a string theory with a target space composed of regular star polygons. This solvable string theory provides an explicit example of a decaying stochastic solution to the unforced Navier-Stokes equation.
- While we have not proven that this solution is reachable from smooth initial data (corresponding to vanishing noise), the presence of thermal noise in physical fluids ensures its relevance in realistic scenarios.
- We have established a No Explosion Theorem, ruling out finite-time blow-ups for arbitrary initial data with finite noise. This result leaves only two viable alternatives: smooth laminar solutions and the decaying turbulence solution (Euler ensemble = discrete string theory).
The insights presented in this work suggest that turbulent flows, despite their apparent randomness, may be governed by universal principles encoded in the geometry of loop space. By connecting turbulence to discrete geometry and solvable string theories, we provide a foundation for new approaches to understanding turbulence across scales. This framework has implications not only for fluid mechanics but also for broader applications in mathematical physics, including magnetohydrodynamics, compressible flows, and flows in nonlinear spaces.
🔗 Read the full preprint on arXiv: https://arxiv.org/html/2411.01389v5