Navier–Stokes Regularity: Where the Difficulty Sits

Global Regularity of 3D Navier–Stokes: An Energy Argument

A conditional bound Ω_A ≤ C E(0)^5/3/ν^4/3 on the active-shell enstrophy under three cascade hypotheses, together with the explicit smooth data for which those hypotheses fail.

Angular Relaxation on the Integer Lattice

An effective PDE for angular-resolved shell energy fitted to DNS in 1D, 2D and 3D, reproducing the 2D data within 2%; the 3D closure is a fitted description of the data, not a derived equation.

Energy Conservation and the 3D Galerkin Solver

The corrected energy-conserving 3D Galerkin solver, the history of its −i defect, and two lattice lemmas on triad geometry; the rigorous content is conditional and bounds nothing about a particular solution.

Unified Spectral Framework for Arithmetic Oscillations

Truncated Odlyzko–te Riele sums with computed ζ′(ρ_n) over 2,001,052 zeros: the Mertens sum crosses 1 at K = 58 and 2 near K = 1,040; zeta zeros recovered by Fourier analysis of M(x)/√x.

Scaffold Array Diagnostics: Euler, SQG & MHD

Galerkin truncations of Euler, MHD and SQG relax to the Lee–Kraichnan absolute equilibrium, which accounts for the observed enstrophy growth; a finite Galerkin system says nothing about blow-up of the three PDEs.

Shell Model, Diagnostic H, First 3D Kernel

A hand-constructed cubic shell model with a multi-level diagnostic H and the first 3D kernel whose mode-enumeration, triad-precomputation and Leray-projection structure became the basis of the corrected solver; no conclusion about 3D Navier–Stokes follows from it.

Unified Adaptation Theorem

Sufficient conditions for convergence of sequentially composed contractive updates on a shared parameter: block contraction with a non-negative coupling matrix plus LaSalle, with trajectory diagnostics B = 1 + 2I and the convergence-score lemma.

Stretching Drops Out of the Direction Equation

The tangential strain forcing on the vorticity direction, P_ξ⊥Sξ, contains no stretching component: α = ξ·Sξ drops out and only the off-diagonal shear rotates the direction. The forcing splits into an alignment part and a Calderón–Zygmund commutator part bounded by the bmo norm of the direction.

What it clarifies: direction dynamics and amplitude dynamics separate; the direction is driven by shear alone.

The Open Step Is a Bounded-Constant Comparison

The Dirichlet energy of the vorticity direction on a ball controls its mean-square oscillation with the first nonzero Neumann eigenvalue, μ_N ≈ 4.333: D_r ≥ (μ_N/r²) W_r. This kinematic Poincaré inequality holds for any field. The direction-oscillation bootstrap closes if and only if C_⋆(κ/ν)‖ξ‖_bmo ≤ μ_N/2, an r-independent condition on the core Reynolds number.

What it clarifies: the remaining step is a constant comparison, not a scaling gap.

Coherence Does Not Stop a Cascade

The Burgers vortex has exactly unidirectional vorticity and sustained stretching α = γ > 0. Aligned vorticity constrains only the self-induced velocity; strain induced by vorticity elsewhere stretches an aligned tube.

What it clarifies: Liouville theorems based on vorticity coherence are closed as a strategy class.

No One-Point Direction Statistic Is Necessary for Stretching

A divergence-free field of six Burgers-type patches has an exactly isotropic direction second moment (‖Q‖ = 0, resultant 0) and enstrophy-weighted stretching 0.999. On DNS at the enstrophy peak the mean-strain term is at most 6% of the stretching at the flux scale; the strain–direction covariance carries it (Betchov's identity in explicit form).

What it clarifies: the difficulty sits in the strain–direction covariance at the flux scale.

A Pointwise Kolmogorov Closure Is Not a Property of Smooth Flows

The hypothesis |Π_K| ≤ αK E_K^3/2 for every shell fails at t = 0 for a datum with a spectral gap (E₁₀ = 0, Π₁₀ = −1.04), for localised vortex blobs of size ℓ at their own scale by a factor of order ℓ⁻³ (13.6× to 55× for ℓ = 0.25 to 0.15), and in the exponential dissipation range of generic data (7.5×, 30×, 98× at K = 40, 48, 52).

What it clarifies: such a closure excludes concentration a priori and cannot be a hypothesis of a regularity theorem.

Uniform Intra-Shell Dispersion Fails for Exact Solutions

For the shear flow u = f(x₁)e₂ every shell has two modes, so the dispersion constant is n_K/2: 9, 31, 175, 625, 2517 at K = 1, 2, 5, 10, 20.

What it clarifies: uniform dispersion cannot be assumed for all smooth solutions; it delimits where the conditional bound applies.

Conditional Active-Shell Enstrophy Bound

Under uniform dispersion, the pointwise closure and cascade contiguity, the enstrophy carried by the active shells satisfies Ω_A ≤ C E(0)^5/3/ν^4/3, and any uniform-in-time enstrophy bound gives regularity by Prodi–Serrin.

What it clarifies: exactly which cascade hypotheses would suffice, and, with the two results above, where they hold.

Aligned Critical Profiles Are the Axisymmetric Family

For homogeneous critical profiles ω = |x|⁻²Ψ(x̂), the angular shear G = P_ξ⊥Σξ vanishes identically (to 10⁻¹⁵) for swirl-free axisymmetric profiles and for nothing else found: the linearisation at four such profiles has null space equal to the symmetry count (l ≤ 4), next singular value 0.7–1.9% of the largest, and six nonlinear minimisations from random profiles converge to that family. A computed result within band-limited profiles.

What it clarifies: Grujić's alignment condition holds for the leading profile only on the axisymmetric family; the residual is a non-symmetric core whose direction is driven at no less than about 1% of κ/|x|² per unit time.

Inviscid Truncations Thermalise

Ω/E reaches the lattice mean ⟨k²⟩ to within 1% at every N from 3 to 8, unchanged under time-step halving. This is the Lee–Kraichnan absolute equilibrium.

What it clarifies: the enstrophy growth observed in truncated Euler is equilibration, not evidence about the PDE.