Analysis and Mathematical Physics

We reduce the solution of the Navier-Stokes (NS) equation in infinite Euclidean space of arbitrary dimension to a one-dimensional singular problem. Assuming rough initial data, characterized by a Gaussian distribution around an arbitrary velocity field with variance $\sigma\to 0$, we argue that such roughness is intrinsic to physical fluids due to thermal fluctuations. This generalization yields a novel representation of the generating functional for the velocity circulation distribution (loop functional), evolving as a momentum loop $\vecP(\theta, t)$ governed by a nonlinear PDE.

The quantum representation redefines the Reynolds number as a property of the initial momentum loop data, and its evolution is described by the universal momentum loop equation (MLE), independent of scale parameters. Two primary asymptotic regimes exist: laminar flow (small circulation-to-viscosity ratio) and decaying turbulence. We rule out the third alternative — finite-time explosion — based on the inconsistency of such a solution to the MLE, regardless of the initial data.