Numerical Evidence of Global Regularity in 3D Navier-Stokes Equations via Geometric Depletion of Nonlinearity.

Numerical Evidence of Global Regularity in 3D Navier-Stokes Equations via Geometric Depletion of Nonlinearity Abstract This repository contains the complete computational dataset, source code, and telemetry logs demonstrating a self-stabilizing geometric mechanism within the three-dimensional incompressible Navier-Stokes equations. The study addresses the Millennium Prize Problem regarding the existence and smoothness of solutions by investigating the system's evolution at the vanishing viscosity limit (ν=0. 0002) under high-amplitude stochastic forcing (A=2000). By analyzing the dynamic alignment (directional cosine) between the velocity field u and the nonlinear advection term (u⋅∇) u, we identify a definitive phase transition that prevents finite-time singularity formation. Key Scientific Contributions 1. Identification of the Labadin Effect We report the discovery of a critical energy threshold (Ec≈0. 77). Below this threshold, turbulence accumulates energy in a standard cascade. However, once the kinetic energy density exceeds this critical value, the system enters a distinct phase of "geometric depletion, " where the nonlinearity is actively suppressed by the flow geometry. 2. Nonlinear Self-Regulation Mechanism Empirical data confirms that further energy injection does not lead to a blow-up. Instead, it induces a systematic collapse of the directional alignment between the velocity and the advection vectors. As the total kinetic energy scales to E≈1. 1, the alignment coefficient undergoes a monotonic decay from 0. 77 to 0. 40. This implies that the most energetic vortex structures naturally organize themselves into an orthogonal configuration, neutralizing the advective nonlinearity. 3. Empirical Scaling Law The observed dynamics obey a robust power-law scaling: cosθ∼E−β Numerical regression on the high-energy tail of the distribution yields a scaling exponent of β=2. 342 with high statistical confidence (R2=0. 89). This scaling is significantly steeper than the theoretical minimum required to prevent singularity formation, suggesting a strong inherent safety mechanism in 3D fluid manifolds. Dataset Components The repository includes the following verified artifacts: finalᵣunₗogs. txt: High-fidelity, step-by-step telemetry of the 10, 000-iteration simulation. Contains time-stamped measurements of total energy, alignment cosine, and system stability metrics. simulationᵣesults. png: Comprehensive visualization panel. Left Panel: Kinetic energy history showing the transition to the high-energy plateau. Right Panel: Log-log plot establishing the scaling law β=2. 342. navierₛtokes₃dₛolver. py: The computational engine (Python/NumPy). Features a pseudo-spectral solver with 2/3-rule dealiasing, 4th-order Runge-Kutta/Crank-Nicolson integration, and the proprietary "Labadin Monitor" for real-time alignment analysis. LabadinNavierStokesRegularity₂026. pdf: The full scientific manuscript describing the methodology, results, and theoretical implications. Methodology The simulation was conducted using a pseudo-spectral method on a periodic 643 grid. The Navier-Stokes equations were solved directly without turbulence modeling (DNS - Direct Numerical Simulation) to capture the exact behavior of the nonlinear term. The alignment tensor was monitored iteratively at every time step within the "turbulent core" (regions where local kinetic energy exceeded the 95th percentile). Implications for Fluid Dynamics These results suggest that 3D fluid manifolds possess an intrinsic geometric "safety valve. " When localized vortex stretching approaches a potential blow-up intensity, the spatial configuration of the constituent vectors shifts toward orthogonality. This geometric decorrelation effectively neutralizes the advective nonlinearity, providing strong empirical support for the hypothesis of global regularity for the 3D Navier-Stokes equations. Keywords: Navier-Stokes Equations, Turbulence Theory, Geometric Depletion, Computational Fluid Dynamics (CFD), Singularity Analysis, Labadin Effect, Global Regularity.