A Regularized Navier-Stokes Model with a Non-Local Gravitational Non-Collapse Principle

A Regularized Navier-Stokes Model with a Non-Local Gravitational Non-Collapse Principle. Jorge Clavijo Hernández - ORCID: 0009-0001-3855-9593 - A Coruña, Spain - July 2026 ABSTRACT: Starting from the empirical observation that nature does not generate vortices of infinite energy — from a swirl in a glass to a hurricane or a spiral galaxy — we conjecture a physical principle of non-collapse. We propose a regularized Navier-Stokes system where concentrated kinetic energy pays a non-local cost G = |u(x)|^2|u(y)|^2 / |x-y| dx dy that diverges as $1/r$ when r → 0. Building on Arnold's geodesic interpretation of the Euler equations (1966) and Perelman's W-functional for Ricci flow (2002), we formulate a coupled system with a corrective term ε > 0. We present formal derivations of monotonicity for the combined functional Φ = WNS + ε G and numerical evidence in reduced 0D vorticity and real 1D non-local Burgers-type models showing gradient saturation and shock prevention for ε > 0. Important Note: This work does NOT claim to analytically solve the classical Millennium Navier-Stokes problem in its pure form. To propose and study a modified regularized system. Author's Note:This work does not arise from a desire to challenge mathematical complexity out of mere intellectual pride, but from a deep contemplation of the harmony within universal design. The persistence and stability of vortices in nature, from the microscopic scale to galactic structures, remind us that the cosmos operates under a perfect order that rejects the chaos of infinite collapse. The limitations of our current mathematical models do not reflect an error in creation, but rather the limit of our own human understanding, as expressed in the Book of Job: "Where were you when I laid the earth's foundation? Tell me, if you have understanding. Who marked off its dimensions? Surely you know!" (Job 38:4-5). This regularized model is a humble attempt to sketch a mathematical map that acknowledges the physical limits inscribed in that original wisdom. Keywords: Regularized Navier-Stokes, non-collapse principle, Perelman W-functional, Arnold geodesic flow, non-local model, Navier-Stokes regularization, entropy monotonicity License: CC-BY 4.0 Code Availability: The 3D pseudo-spectral simulation code (Python / NumPy FFT) used to generate the 32³ results is available upon private request for replication and scaling to higher resolutions ( 64³, 128³, 512³). Contact: Clavi20000@gmail.com