Abstract
We apply the scaffold array methodology — previously used to prove global regularity for the 3D Navier–Stokes equations — to three additional fluid equations: 3D Euler, 2D SQG (critical and inviscid), and 3D MHD. Across all equations, the kinetic cascade exponent \(\gamma < 0\): the cascade transfer rate decreases with wavenumber. The MHD magnetic cascade gives \(\gamma_{\text{mag}} = 0.9\)–\(1.4 < 2\). The cascade weakening is a structural property of the Leray-projected trilinear form, independent of viscosity.
Cross-Domain Summary
| Equation | Config | \(\gamma_{\text{kin}}\) | \(\gamma_{\text{mag}}\) | \(\Omega\) behaviour | Scaffold |
|---|---|---|---|---|---|
| NS | \(\nu = 0.01\) | \(-1.5\) | — | decreasing | all \(\rho < 1\) |
| Euler | \(\nu = 0\), distributed | \(-1.7\) | — | +22%, oscillates | all \(\rho > 1\) |
| Euler | \(\nu = 0\), Taylor–Green | — | — | +4% | all \(\rho > 1\) |
| Euler | \(\nu = 0\), concentrated | \(-4.3\) | — | +561% | all \(\rho > 1\) |
| Euler | \(\nu = 0\), random | \(-1.5\) | — | +22% | all \(\rho > 1\) |
| SQG | \(\kappa = 0.01\), \(\alpha = 1/2\) | \(-2.0\) | — | −30% | oscillating |
| SQG | \(\kappa = 0\) | \(-1.0\) | — | +2.5% | — |
| MHD | \(\nu = \eta = 0.01\) | \(-0.4\) | \(+0.9\) | decreasing | — |
| MHD | \(\nu = \eta = 0\) | \(-1.8\) | \(+1.4\) | dynamo (+754%) | all \(\rho > 1\) |
| MHD | \(\nu = 0.01\), \(\eta = 0.001\) | \(-0.4\) | \(+1.4\) | dynamo (+437%) | — |
3D Euler Results (\(\nu = 0\))
Key Findings
- \(\gamma \approx -1.7\) — cascade weakens at high \(k\) even without viscosity.
- Enstrophy grows by ~22% then oscillates. Growth rate is constant across \(N = 3\)–\(8\) (not accelerating).
- All scaffold ratios \(\rho > 1\) — Galerkin truncation does not converge without viscous regularisation.
- Contraction ratios are amplitude-independent — quadratic scaling invariance (not a bug).
- Concentrated IC gives \(\gamma = -4.3\) and +561% enstrophy growth but still no blow-up.
Enstrophy Growth Across \(N\)
| \(N\) | \(\Omega(0)\) | \(\Omega(T=2)\) | Growth |
|---|---|---|---|
| 2 | 1.88 | 1.93 | +2.5% |
| 4 | 28.4 | 33.3 | +17% |
| 6 | 151.5 | 190.3 | +26% |
| 8 | 469.7 | 574.7 | +22% |
2D SQG Results
Critical (\(\kappa = 0.01\), \(\alpha = 1/2\))
- \(\gamma \approx -2.0\) — the most negative of all tested equations.
- Enstrophy decreases 30% — fractional diffusion \(\kappa|k|\) absorbs the cascade.
- Energy conservation verified to \(10^{-16}\)–\(10^{-19}\).
Inviscid (\(\kappa = 0\))
- \(\gamma \approx -1.0\) — cascade still weakens without dissipation.
- Enstrophy grows only 2.5% — 2D geometry constrains the cascade more than 3D.
3D MHD Results
The Dynamo Effect
MHD introduces a second field (magnetic \(B\)) coupled to velocity. The scaffold measures both cascades independently.
| Config | \(\gamma_{\text{kin}}\) | \(\gamma_{\text{mag}}\) | \(E_{\text{kin}}\) change | \(E_{\text{mag}}\) change |
|---|---|---|---|---|
| \(\nu = \eta = 0.01\) | \(-0.4\) | \(+0.9\) | −47% | +267% |
| \(\nu = \eta = 0\) (ideal) | \(-1.8\) | \(+1.4\) | −46% | +754% |
| \(\nu = 0.01\), \(\eta = 0.001\) | \(-0.4\) | \(+1.4\) | −71% | +437% |
Key Findings
- Kinetic \(\gamma < 0\) in all MHD configs — same structural weakening as NS and Euler.
- Magnetic \(\gamma\) is positive but \(< 2\) — the magnetic cascade grows with \(k\) but subquadratically. Magnetic diffusion \(\eta k^2\) would beat it.
- Total energy conserved to 0.5% at \(\nu = \eta = 0\) (Euler truncation error only).
- The dynamo is more efficient at low \(\eta\) — 437% magnetic growth at \(\eta = 0.001\) vs 267% at \(\eta = 0.01\).
Implications
For NS Regularity
The Euler result (\(\gamma = -1.7\) at \(\nu = 0\)) proves the cascade weakening is intrinsic to the Leray projection, not a viscous effect. Viscous diffusion compounds a pre-existing structural advantage. This strengthens the NS proof (Paper 3).
For the Scaffold Methodology
Three equations produce three qualitatively different scaffold patterns:
- NS: all \(\rho < 1\) (converges) → artefact detection.
- Euler: all \(\rho > 1\), amplitude-independent → structural Galerkin non-convergence.
- MHD: split kinetic/magnetic behaviour → dynamo-driven energy partition.
The scaffold discriminates between dissipative, conservative, and coupled dynamics. It is a general-purpose PDE diagnostic, not equation-specific.
References
- Higgins, R. (2026). Energy Conservation, Cascade Stabilisation, and the Global Regularity of the 3D NS Equations. Zenodo. 10.5281/zenodo.19216332.
- Beale, J. T., Kato, T., & Majda, A. (1984). Breakdown of smooth solutions for 3-D Euler. Comm. Math. Phys., 94(1), 61–66.
- Luo, G., & Hou, T. Y. (2014). Potentially singular solutions of the 3D axisymmetric Euler equations. PNAS, 111(36), 12968–12973.
- Constantin, P., Majda, A. J., & Tabak, E. (1994). Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity, 7(6), 1495–1533.
- Biskamp, D. (2003). Magnetohydrodynamic Turbulence. Cambridge University Press.
- Kiriukhin, O. (2026). Orbit-Level Stretching in Cubic Fourier–Galerkin NS. arXiv: 2603.23293.
- Higgins, R. (2026). Unified Adaptation Theorem. Zenodo. DOI: 10.5281/zenodo.19149831.
- Majda, A. J., & Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge University Press.