What question did this study set out to answer?

The research aims to demonstrate that Leray–Hopf weak solutions of the Navier–Stokes equations remain smooth for all time under specific initial conditions.

May 1, 2026Open Access

Phase Decoherence and Regularity of the Three-Dimensional Navier–Stokes Equations

Key Points

The research aims to demonstrate that Leray–Hopf weak solutions of the Navier–Stokes equations remain smooth for all time under specific initial conditions.
Proof focuses on arithmetic structure in nonlinear terms of the equations.
Uses helical Fourier basis to analyze velocity fields and introduce phase decoherence mechanisms.
Applies the Beale–Kato–Majda criterion to establish global regularity.
Shows that distinct modes exhibit phase independence due to different triadic partners' forcing.
Demonstrates global regularity for all time under the conditions specified for the initial data.
Introduces a deterministic contraction map driving phase coherence across spectral shells.

Abstract

We prove that any Leray–Hopf weak solution of the incompressible Navier–Stokes equations on the periodic torus T³ = (R/2πZ) ³ with viscosity ν > 0 and zero external forcing remains smooth for all time, provided the initial datum u₀ ∈ H¹ (T³) is divergence-free. The proof exploits arithmetic structure in the nonlinear term B (u, u) that is invisible to harmonic analysis estimates alone, precisely the "finer structure" identified by Tao (2016) as necessary after his finite-time blowup construction for an averaged Navier–Stokes equation. The argument is genuinely three-dimensional: SO (3) non-commutativity makes the triadic coupling coefficient complex-valued, so the coherent state is not a stable attractor. The mechanism has no analogue in dimension two, where the real-valued coupling permits the inverse cascade that sustains coherence. The proof introduces a phase decoherence mechanism. Expanding the velocity field in a helical Fourier basis, we show that the phases of individual modes behave as weakly coupled oscillators on each spectral shell. The Leray projector and the arithmetic of the integer lattice Z³ together ensure that distinct modes receive forcing from disjoint sets of triadic partners, making their phase evolution conditionally independent. A deterministic contraction map then drives the phase coherence on every shell to O (1), uniformly in the shell index and for every initial configuration. Combined with a new Uniform Shifted Coherence theorem and the Beale–Kato–Majda criterion, this yields global regularity. Changes in v1. 5 (2026-04-29): Single-path restructure. Bootstrap proof of χK = O (1) is the single load-bearing route; independent routes (GRH-conditional, Bessel/DPI, tower property) moved to a companion. §6 reformulated as a Stability Dichotomy via the Pythagorean identity. Adler-to-Giacomin reduction made explicit (Proposition). Bessel-contraction proof corrected (Jacobi-Anger identity). Changes in v1. 4 (2026-04-24): Star Invariant construction (Def 10. 4) rewritten as an honest algebraic cluster-graph model, with alphabet-cage alignment remark deriving 52 = 4×13. Second unconditional proof route added via the Inverse Cascade Trap. Trace-zero proposition closes the Tr (J) =0 gap. CLT distributional sketch removed (replaced with an open-problem remark). Notation and citation hygiene tightened paper-wide. Changelog updated. Changes in v1. 3 (2026-03-30): Tao comparison response (§1. 2), proof-chain tightening (44pp, down from 48), Star Invariant proof corrected (direct algebraic computation replacing Fiedler propagation), 45 editorial fixes including conditional independence made explicit, K-uniformity corrected, numerical language hardened, and speculative appendices quarantined. Full changelog added: changelog. pdf Changes in v1. 2 (2026-03-29): Seven additive changes strengthening analytical rigor. Changes in v1. 1 (2026-03-27): Corrected 6j-symbol eigenvalue claim in §E. 5.

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