Global Regularity of a Dynamic Coherence Closure of the 3D Navier–Stokes Equations

The three-dimensional Navier–Stokes equations (NSE) remain mathematically incomplete: no proof of global regularity exists for the standard system without additional structural hypotheses. We argue that a key missing ingredient is a thermodynamic feedback mechanism encoding the self-organisation of turbulent energy flow. We introduce the Dynamic Coherence Closure (DCC) framework (κ = −1 regime), which augments the NSE with a dynamically evolving coherence state variable C ∈ (0, 1) governed by a logistic equation driven by cumulativeenergy dissipation. This coupling yields a coherence-mediated, Ladyzhenskaya-type viscosity νC = αC (1 −C) |∇u| that naturally self-regulates. Under two structural assumptions—spatial homogeneity of the coherence field and a coercive estimate for the nonlinear DCC operator (established in the appendix via a direct p-Laplacian monotonicity argument for the full-gradient operator) —we prove global regularity of strong solutions for the DCC–NSE system in the force-free case f ≡0, which is an essential restriction of the present analysis. The proof relies on a strict physical bound on cumulative energy dissipation, which locks C away from the singularity C = 1. This persistent nonlinear damping absorbs the super-linear H1 growth of the convective term, precluding finite-time blow-up. Finally, a parameter-free statistical consistency check against published direct numericalsimulation (DNS) data (Re_λ ≈ 140–1000) shows that real turbulence possesses precisely the dissipation statistics required for the DCC auto-regulatory mechanism: the observed plateau CVε ≈1 yields the parameter-free prediction σlnC ≈ 0. 833 for a coherence-type field, consistent with DNS dissipation statistics