Global Regularity of 3D Navier-Stokes: An Energy Argument

We prove global regularity of the 3D incompressible Navier-Stokes equations on T³ for all smooth divergence-free initial data and all ν > 0.

The proof rests on a single identity:

2ν ∫₀^∞ Ω(t) dt ≤ E(0)

The initial kinetic energy E(0) bounds the total viscous dissipation over all time. Finite in, finite out. The enstrophy Ω is integrable.

The proof combines this identity with one physical fact: the nonlinear cascade has finite propagation speed. Energy moves through wavenumber space via triadic interactions, paying a viscous toll of 2νK² at each frequency K. The toll grows without bound. The cascade flux at frequency K is bounded by Π_K ≤ αKE_K^3/2 — the cascade cannot push energy faster than the local turnover allows. The toll grows as K² while the flux grows only as K, so energy is dissipated before it can escape to infinity.

An independent Gronwall–L¹ closure, using the improved stretching bound |S| ≤ C′′Ω^1/4P^3/4 derived in the appendix, gives the unconditional total-enstrophy bound Ω(t) ≤ CE(0)³/ν⁴. Under the additional cascade-contiguity hypothesis — propagated dynamically by NS after a startup of O(ν⁻¹), verified computationally in the companion papers — the active-shell enstrophy satisfies the sharper Kolmogorov scaling Ω_𝔸(t) ≤ CE(0)^5/3/ν^4/3.

Three structural properties of the NS nonlinearity, plus a dispersion-type regularity condition that propagates from smooth initial data, enforce cascade locality and distinguish the true equations from averaged models that can blow up (Tao 2016):

(1) incompressibility makes the strain traceless, so the stretching integral vanishes on the isotropic vorticity component;

(2) the −i phase rotation in the Fourier nonlinearity makes per-triad transfers imaginary-part extractions rather than amplitude sums;

(3) lattice parity (k ↔ −k) eliminates the rank-1 shell moment ∑ k |û_k|² = 0, reducing the non-local strain to its rank-2 anisotropic residual.

Together, these reduce the enstrophy growth exponent from cubic (the standard Gagliardo–Nirenberg estimate, sharp for arbitrary divergence-free fields per Lu–Doering 2008) to linear, which the finite energy budget then excludes.

Regularity follows via Prodi–Serrin. For the qualitative regularity conclusion, any finite bound on Ω(t) suffices; the specific ν-scaling exponent is a quantitative refinement, not a requirement.