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This work explores the mechanisms ensuring the regularity of the 3D Navier-Stokes equations, preventing blow-ups in finite time.

April 12, 2026Open Access

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

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This work explores the mechanisms ensuring the regularity of the 3D Navier-Stokes equations, preventing blow-ups in finite time.
Identification of geometric frustration in the trilinear form on the lattice
Proof of the Lattice–Leray and Global Frustration Lemmas
Development of a 3D Galerkin solver for exact Fourier trilinear coupling
Validation through nine independent methods across three programming languages
Non-vanishing angular variance in triad interactions observed
Each Fourier mode linked to multiple conflicting triads, indicating geometric constraints
Phase cancellation significantly mitigates blow-up potential, observing 97-99.9% cancellation
Computational methods confirm energy conservation and cascade subcriticality across various environments

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Findings demonstrate stability of the 3D Navier-Stokes equations, highlighting implications for fluid dynamics.

Rod Higgins (2026) studied this question.

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