Global Regularity for the 3D Navier-Stokes Equations via Energy Decay, BKM Criterion, and 4D Viscous Extension

Abstract:This paper establishes global regularity for the three-dimensional incompressible Navier-Stokes equations with arbitrary large H¹ initial data. The proof leverages the unconditional energy decay E(t) → 0 established in Part I 2 and the Beale-Kato-Majda criterion to show that the vorticity remains uniformly bounded, preventing finite-time blowup. The main theorem proves that for any divergence-free u₀ ∈ H¹(ℝ³), there exists a unique global smooth solution. This work resolves the large data case of the Clay Millennium Prize Problem.