Energy Conservation, Cascade Stabilisation, and the Global Regularity of the 3D Navier–Stokes Equations

Rod Higgins · Senuamedia · v19, September 2026 · DOI: 10.5281/zenodo.22710239

Abstract

Current version: v19 · DOI 10.5281/zenodo.22710239

Contents

1. Introduction
2. The 3D Galerkin Framework
3. The Energy Conservation Defect
4. The Corrected Solver
5. Independent Validation
6. Lattice Geometry
7. Scaffold Array Diagnostics
8. Parameter Sweeps
9. Discussion

1. Introduction

The Problem

The 3D incompressible Navier–Stokes equations on the periodic torus \(\mathbb{T}^3\):

The Clay Millennium Prize Problem asks: given smooth initial data, does the solution remain smooth for all time? This paper does not answer that question. It records the computational infrastructure built to study the Galerkin truncations, what the truncations were measured to do, and two facts about the triad geometry of the lattice.

What This Paper Contains

1. A 3D Fourier–Galerkin solver with exact trilinear coupling, and the discovery and correction (2026-03-24) of a missing \(-i\) factor in its first, real-coefficient implementation, which injected up to 15% of the energy at \(N = 7\).
2. Independent validation of the corrected kernel: a NumPy implementation, scipy RK45, the Taylor–Green exact decay, and an FFT pseudo-spectral solver that agrees with the triad kernel to \(2.8 \times 10^{-17}\) at \(N = 4\) over \(T = 0.01\).
3. Two lattice facts. The angular variance \(\mathrm{Var}(\cos^2\alpha_T)\) of triad interactions on \(\mathbb{Z}^3\) is positive at every shell computed (\(\ge 0.085\) through \(K = 55\)) and tends to \(4/45\); and modes shared by several triads cannot maximise the stretching of all of them at once. Neither bounds anything about a particular solution.
4. Diagnostics and measurements: scaffold arrays across nested truncation levels, weighted enstrophy, and parameter sweeps, reported as data.
5. The conditional statement: a uniform-in-\(N\) enstrophy bound for the truncations implies smoothness of the limit via the Galerkin limit and the Prodi–Serrin criterion. No such bound is established.

2. The 3D Galerkin Framework

Fourier-Space NS Equation

where \(P_{\mathbf{k}}(v) = v - (\mathbf{k} \cdot v / |\mathbf{k}|^2)\mathbf{k}\) is the Leray projection. The factor \(-i\) is essential for energy conservation.

Energy Identity

This holds because the advection operator is skew-symmetric with respect to the \(L^2\) inner product when the Leray projection and the \(-i\) phase factor are correctly applied.

The Conditional Statement

If the enstrophy of the Galerkin solutions \(u^{(N)}\) is bounded uniformly in \(N\) and in time, the Galerkin limit lies in \(L^\infty(H^1) \hookrightarrow L^\infty(L^6)\) and is smooth by the Prodi–Serrin criterion. This is classical. The paper asserts no such bound: the standard estimate \(|S| \leq C_S \Omega^{1/4} \Sigma^{3/4}\) for the stretching term is sharp for arbitrary divergence-free fields, and a finite Galerkin system has bounded enstrophy for trivial reasons, so measurements on the truncations do not supply the bound.

Mode Counts

3. The Energy Conservation Defect

The v2 solver computed:

Missing the \(-i\) factor. The correct expression is:

Energy Drift at \(\nu = 0\) (v2 Solver)

The drift is independent of \(\Delta t\), which places the error in the right-hand-side computation rather than the integrator. The defect was found and corrected on 2026-03-24.

4. The Corrected Solver (v3)

Complex Fourier coefficients (6 doubles per mode) with \(-i\) applied correctly:

v2 vs v3 Energy Change at \(\nu = 0.01\)

5. Independent Validation

With \(\nu = 0\), the truncated systems thermalise: \(\Omega/E\) reaches the lattice mean \(\langle k^2\rangle\) to within 1% at every \(N\) from 3 to 8, unchanged under time-step halving. This is the Lee–Kraichnan absolute equilibrium and accounts for the enstrophy growth observed in truncated Euler runs.

6. Lattice Geometry

Two facts about the triadic interactions \(\mathbf{p} + \mathbf{q} = \mathbf{k}\) on \(\mathbb{Z}^3\), computed exhaustively shell by shell (experiment_leray_angle_variance.c, experiment_triad_conflict.c).

Both statements are about the lattice and the coupling coefficients only. They constrain the form of the nonlinearity; they do not bound the enstrophy of any particular solution, and no such bound is drawn from them here.

7. Scaffold Array Diagnostics

The scaffold array integrates the same datum at nested truncation levels and records, for each diagnostic (energy \(E\), enstrophy \(\Omega\), shell profile \(\eta\), and the higher-level composite \(H'''\)), the ratio \(\rho\) of successive between-level differences. The table reports the ratios measured with the corrected kernel at four initial amplitudes.

The ratios are reported as measured. They describe how the diagnostics change between truncation levels on a fixed window and are not used here as evidence of convergence to a solution of the equations. The diagnostic's practical role was to expose the \(-i\) defect: with the first kernel the enstrophy at \(A = 0.30\) diverged, and the divergence was produced entirely by the solver.

8. Parameter Sweeps

16 configurations on 7 parallel AWS EC2 nodes, spanning 3 orders of magnitude in \(\nu\), 1 order in \(A\), and 4 initial-condition families. For each run the per-shell transfer \(T_K\) was fitted to the form \(C_{\mathrm{L}}\,E\,\Omega^{1/2}\) with prefactor \(C_{\mathrm{L}}\), and the stretching term to \(C_{\mathrm{s}}\,\Omega^{3/2}\). The fitted prefactors and the peak enstrophy are reported as data; they are finite-window, finite-\(N\) measurements of the truncated system.

Viscosity × Amplitude Sweep (Distributed IC)

Initial Condition Sweep

Observations

1. At \(A = 0.1\) the stretching prefactor \(C_{\mathrm{s}} = 0.041\) is the same to three significant figures at \(\nu = 10^{-2}\), \(10^{-3}\) and \(10^{-5}\).
2. For the distributed and random families \(C_{\mathrm{L}} \in [0.016, 0.026]\); the Taylor–Green and concentrated families give prefactors up to ten times larger.
3. \(C_{\mathrm{s}}\) scales approximately as \(A^{0.9}\) over the range tested.
4. All values are for the truncated system on a fixed window; a finite Galerkin system cannot blow up, and these measurements are not evidence about the partial differential equation.

9. Discussion

What stands

1. The corrected kernel. Complex coefficients with the \(-i\) factor conserve energy exactly under the nonlinearity; the earlier real-coefficient implementation injected up to 15% at \(N = 7\). Three implementations and an FFT pseudo-spectral solver agree.
2. The two lattice lemmas on triad geometry (Section 6), which are statements about \(\mathbb{Z}^3\) and the Leray projection.
3. The conditional statement that a uniform-in-\(N\) enstrophy bound implies smoothness.

What the measurements do not show

A finite Galerkin system has bounded enstrophy for trivial reasons, and its inviscid truncations thermalise to the Lee–Kraichnan equilibrium. Finite-window excursion amplitudes depend on the window (\(\propto T^{-0.87}\)), the threshold and the solver, and are not regularity thresholds. Tao (2016) shows that an averaged equation with the same energy identity blows up, so an argument for regularity must use structure of the actual nonlinearity beyond energy conservation. Where that structure sits, on current evidence, is in the covariance between strain and vorticity direction at the flux scale and in concentration; the companion direction-identity and coercivity notes address that question directly.

Reproducibility

All code at github.com/senuamedia/lab-code: the v3 kernel (solvers/v3/triad_kernel_v3.c), the validation scripts (validation/), and the lattice experiments (ns-proof-experiments/). The full validation set runs in under 30 minutes on a laptop.