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Energy Conservation, Cascade Stabilisation, and the Regularity of the 3D Navier-Stokes Equations: Computational Evidence from Galerkin Truncations

Authors: Higgins, Rod;

Energy Conservation, Cascade Stabilisation, and the Regularity of the 3D Navier-Stokes Equations: Computational Evidence from Galerkin Truncations

Abstract

We present computational evidence for the global regularity of the three-dimensional incompressible Navier–Stokes equations on the periodic torus T 3 = (R/2πZ) 3 Through the development of a multi-perspective scaffold array methodology — which measures the same Galerkin system from multiple truncation-level perspectives simultaneously — we discovered that the 3D spectral solver used in our investigation (and potentially in other spectral NS implementations) failed to conserve energy due to a missing imaginary factor −i in the Fourier-space trilinear coupling. This energy conservation failure caused spurious energy injection of 1–15% per unit time (completely independent of the time step ∆t), producing enstrophy growth that was indistinguishable from genuine cascade blow-up. We correct this error by implementing complex Fourier coefficients with the full −i factor, achieving exact energy conservation: P k Re(uˆk · NLk) = 0 to machine precision at every truncation level. Three independent implementations (C, Python/NumPy, and scipy RK45) validate this result: initial energies agree to all digits, evolved energies agree to 9 × 10−6 relative, and the Taylor–Green vortex analytical solution is reproduced to 10−7 . With the corrected solver, we observe that: The forward energy cascade stabilises at a finite wavenumber (N ≤ 14) for all tested initial conditions, with total energy monotonically decreasing and enstrophy bounded. All scaffold array contraction ratios satisfy ρ 0, and we claim that the energy conservation identity — when correctly implemented — is the structural property that prevents blow-up. Previous computational studies that did not verify energy conservation at ν = 0 may have been observing solver artefacts rather than genuine Navier–Stokes dynamics.

Keywords

energy conservation, regularity, millennium prize problem, Navier-Stokes, Leray projection, scaffold array, computational fluid dynamics, Galerkin truncation, cascade stabilisation

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
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