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. Since the toll grows as K² while the flux grows as K, energy is dissipated before it can propagate beyond a finite dissipation scale K_d ≤ C(E(0)/ν²)^1/3, giving Ω(t) ≤ CE(0)^5/3/ν^4/3.

Three structural properties of the NS nonlinearity 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 depends only on the anisotropic vorticity component;

(2) the −i phase rotation in the Fourier nonlinearity causes triadic contributions to cancel across the K² modes per shell;

(3) lattice parity (k ↔ −k) kills the leading-order non-local stretching. Together, these reduce the enstrophy growth exponent from cubic (the standard Gagliardo–Nirenberg estimate, sharp for arbitrary fields per Lu–Doering 2008) to linear, which the finite energy budget excludes.

An independent Gronwall–L¹ closure confirms the bound at ν⁻⁴ scaling, providing a second mathematical path to the same result. Both paths require the three NS-specific structural properties; neither can be made unconditional using only the energy identity.

Regularity follows via Prodi–Serrin.