TITLE:
Geometric Regularization and Internal Frequency Fields in the 3D Navier-Stokes Flow
AUTHORS:
Narcis Petenchia
KEYWORDS:
Navier-Stokes Equations, Regularity, Internal Geometry, Pressure-Frequency Coupling, Graduated Viscosity, Anisotropic Stress, Vorticity Coherence, Enstrophy Inequality
JOURNAL NAME:
Journal of High Energy Physics, Gravitation and Cosmology,
Vol.12 No.1,
December
31,
2025
ABSTRACT: 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 μeff(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
d/
dt
‖ ω ‖
2
+2
μ
min
‖
∇ω ‖
2
+2c
‖
(
b⋅∇
)ω ‖
2
≤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
∗
→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 (αμ, χ, η) → 0, then the limit solution satisfies the classical 3D NSE smoothly for all finite times. The key analytical challenge is maintaining a uniform BKM bound independent of the anisotropic regularization coefficients—a property conjectured to hold due to the intrinsic geometric coupling ∇Ω ∝ −∇p, which remains non-vanishing even in the limit. Physically, the mechanism can be interpreted as an internal redistribution of pressure energy along geometric flow directions. In hydrostatic or stratified configurations, this coupling enforces vertical coherence and horizontal energy dispersion, mirroring the natural stabilization observed in buoyant fluids and microgravity water spheres. The framework thus bridges classical fluid dynamics and geometric field theory, revealing how the internal geometry of pressure and frequency ensures smoothness and finite enstrophy in three-dimensional incompressible flows.