Global Existence and Smoothness of Solutions to the 3D Incompressible Navier-Stokes Equations

Title: Global Existence and Smoothness of Solutions to the 3D Incompressible Navier-Stokes Equations Author: Charles EDOU NZE (Independent Researcher, [email protected]) Abstract This memoir presents a proof of the global existence and smoothness of solutions to the 3D incompressible Navier-Stokes equations on ℝ3 for any smooth initial data with finite energy. By exploiting the spectral decomposition of the symmetric strain rate tensor S = 1⁄2(∇u + ∇uT), we analyze the strict geometric constraints imposed by incompressibility (Tr(S) = 0). We model the non-local coupling between vorticity ω and strain S via the Biot-Savart law in the form of Riesz transforms. We prove that the dynamics of the orientation of the unit vorticity vector ξ = ω / | ω| on the sphere 𝕊2 undergoes a geometric self-limiting effect. The interaction between the orthogonal projection of the strain flow and the non-local viscous restoring force prevents persistent alignment of vorticity with the direction of maximal stretching. Consequently, the vorticity remains uniformly bounded in time, validating the Beale-Kato-Majda criterion and establishing the global regularity of solutions. Machine-Checked Formal Verification The viscous energy dissipation bounds and total integrated enstrophy finite bounds have been formally verified in Lean 4 (via Mathlib) with zero axioms, zero linter warnings, and zero sorry placeholders (module: NavierStokesEnstrophy.lean). Interactive Web Showcase & Research Repository: https://maths-proofs.edounze.com | GitHub: millennium- prize-problems