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May 20, 2026Open Access

Spectral Non-Concentration Criteria for Navier–Stokes Regularity on T³

Key Points

The aim is to demonstrate global regularity for 3D incompressible Navier–Stokes equations on the torus T³ using the Spectral Non-Concentration framework.
Establishes spectral non-concentration criteria for energy distribution.
Uses triad transfer bound and dissipation rates from Littlewood–Paley shell structure.
Applies Gronwall inequality and operator continuity lemma for proof.
Proves normalized shell energy distribution is bounded uniformly (‖a(t) − μ‖_ℓ¹ ≤ 0.039 for all t ≥ 0).
Finds explicit constants: κ, K, and N_eff ≥ 654.
Establishes global regularity on T³ under the SND assumption.

Abstract

We establish a conditional proof of global regularity for the 3D incompressible Navier–Stokes equations on the torus T³ via the Spectral Non-Concentration (SND) framework. The central result, Lemma 2 (SND Simplex Stability), proves that under the SND assumption the normalized shell energy distribution satisfies ‖a (t) − μ‖_ℓ¹ ≤ 0. 039 uniformly for all t ≥ 0, with explicit constants κ, K, and Nₑff ≥ 654. The proof combines a triad transfer bound, an explicit dissipation rate derived from the Littlewood–Paley shell structure, an arithmetic closure via the GCD interaction matrix, and a Gronwall inequality closing the transient regime. Combined with the frozen spectral gap δ₀ = 0. 20 and the operator continuity lemma, global regularity on T³ follows conditionally on SND. This paper is the second in a series; related records: DOI 10. 5281/zenodo. 19842060 and DOI 10. 5281/zenodo. 19842061.

Spectral Non-Concentration Criteria for Navier–Stokes Regularity on T³

Key Points

Abstract

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