Geometric Regularization and Internal Frequency Fields in the 3D Navier-Stokes Flow

We present a geometric framework for the regularization of the three-dimensional incompressible Navier-Stokes equations (NSE) based on the coupling between internal frequency fields and the pressure gradient. The model introduces an effective graduated viscosity μ_eff(p, Ω) and two anisotropic constitutive stresses, σ^(Ω) and σ^[Ω], linked to an internal frequency field Ω(x, t). The constitutive law ？Ω ∝ ？？p establishes a coherent alignment between frequency and pressure, generating an intrinsic geometric damping in the direction most susceptible to vortex stretching. Within this setting, the extended Navier-Stokes system preserves Galilean invariance and classical energy dissipation while producing a coercive enstrophy inequality of the form$\text{d}/\text{d}t{\Vert \omega \Vert }^2+2{\mu }_{\min }{\Vert ？\omega \Vert }^2+2c{\Vert \left(b？？\right)\omega \Vert }^2\le \text{nonlinear？terms}$ , where b = ？Ω/|？Ω| and c_ represents the anisotropic damping amplitude. This new term acts as a directional energy sink, dynamically aligned with ？p, and effectively suppresses local vorticity amplification even when $c_{？}\to 0$ . A global existence theorem (Theorem 8.1) is established for fixed constitutive parameters, followed by a vanishing-regularization program (Theorem 8.2) showing that if uniform Beale-Kato-Majda (BKM) bounds persist as (α_μ, χ, η) → 0, then the limit solution satisfies the classical 3D NSE