A Fractal Unification Framework for the Navier-Stokes Equations Existence Proof, Three-Dimensional Convergence, and Precision Numerical Implementation

his paper proposes a novel framework based on fractal geometry, systemati?cally addressing the existence, smoothness, and high-Reynolds-number turbulencesimulation challenges of the Navier-Stokes equations. By constructing a fractalmanifold MDHregulated by the Hausdorff dimension DH, the existence of weak so?lutions is rigorously proven in the fractal Sobolev space using the Galerkin method.Furthermore, a convergence theorem for fractal solutions in three-dimensional Eu?clidean space is proposed to align with the original formulation of the MillenniumPrize Problem. On the numerical front, an improved fractal SIMPLE algorithmis developed, utilizing dynamic fractal mesh generation and fractal-modified Pois?son equations to achieve efficient turbulence simulations. The numerical analysisdemonstrates:• Performance improvements of the fractal SIMPLE algorithm: Con?vergence speed increased by 40%, with a 15% reduction in error.• High-Reynolds-number turbulence stability: Stable simulations achievedat Re = 106 without oscillations.This work provides a mathematically rigorous and practically implementable solu?tion to the multi-scale turbulence problem and offers a new pathway for addressingthe Millennium Prize Problem.