Angular Relaxation on the Integer Lattice and Global Regularity of 3D Navier–Stokes

Rod Higgins · Senuamedia · v2, September 2026 · DOI: 10.5281/zenodo.22710240

Abstract

Current version: v2 · DOI 10.5281/zenodo.22710240

Contents

1. Introduction
2. The Galerkin Limit: the Conditional Statement
3. The Effective PDE as a Fitted Closure
4. Numerical Results
5. Exact Solutions and Scope
6. Discussion
7. Conclusion
8. References

1. Introduction

The global regularity problem for the 3D Navier–Stokes equations on \(\mathbb{T}^3\) reduces to a single quantity: if the enstrophy \(\Omega(t) = \frac{1}{2}\|\omega(t)\|_{L^2}^2\) is bounded for all time, then \(u \in L^\infty(H^1) \hookrightarrow L^\infty(L^6)\) by Sobolev embedding, and the solution lies in a Prodi–Serrin regularity class. The difficulty is the vortex-stretching term \(\omega \cdot \nabla u\): in 2D it vanishes identically and regularity has been known since Leray (1934) and Ladyzhenskaya (1969); in 3D it is what could drive \(\Omega\) to infinity.

1.1 The angular-resolved observable

The nonlinearity, acting through triadic interactions on the integer lattice \(\mathbb{Z}^3\), moves energy between shells \(K = |\mathbf{k}|\) and also redistributes it among directions within a shell. To see both processes we work with the angular-resolved shell energy

rather than the scalar shell energy \(E_K = \sum_\mu E(K,\mu)\), and we track the shell enstrophy \(\Omega_K\) as an independent state variable.

1.2 Two lattice facts

Both are statements about the lattice and the form of the nonlinearity. Neither bounds \(A_K\) or \(E_K\) for a solution, and Section 5 gives an exact solution for which \(\sqrt{E_K A_K}\) is unbounded.

1.3 Method

1. Run direct numerical simulations: Burgers on \(\mathbb{T}^1\), Navier–Stokes on \(\mathbb{T}^2\), and Navier–Stokes on \(\mathbb{T}^3\), with matched families of initial conditions.
2. Record \(E(K, \mu_j, t)\) and \(\Omega_K(t)\) as independent observables.
3. Fit one family of effective PDEs to all three datasets by minimising the forward-integration error, extracting dimension-dependent coefficients.
4. Use the 2D fit, where the answer is known, as a control: it should return a vanishing stretching coefficient without that being imposed.
5. Report the 3D fits as a description of the data, together with their fit quality and their dependence on \(N\).

1.4 Relation to the companion paper

The companion paper [1] describes the 3D Galerkin solver, its energy-conservation correction, and two further lattice facts about triad geometry. The present paper uses the same solver and adds the angular-resolved observables and the fitted closure.

2. The Galerkin Limit: the Conditional Statement

Compactness. The uniform bounds \(\|u^{(N)}\|_{L^\infty(L^2)} \leq \sqrt{2E(0)}\) and \(\|u^{(N)}\|_{L^\infty(H^1)} \leq \sqrt{2\Omega_{\max}}\) give, by the Aubin–Lions lemma, a subsequence converging weakly in \(L^2(0,T; H^1)\) and strongly in \(L^2(0,T; L^2)\) to a Leray–Hopf weak solution \(u\). Passage of the bound. By weak lower semicontinuity, \(\Omega(t) \leq \liminf \Omega^{(N)}(t) \leq \Omega_{\max}\). Regularity. \(u \in L^\infty(H^1) \hookrightarrow L^\infty(L^6)\) is the Prodi–Serrin class \((p,q) = (\infty, 6)\), so \(u\) is smooth and weak–strong uniqueness gives convergence of the whole sequence.

The hypothesis is the whole problem. A finite Galerkin system has bounded enstrophy for trivial reasons (it is a finite-dimensional ODE with decreasing energy), so bounded \(\Omega^{(N)}\) at any fixed \(N\) says nothing; what is needed is a bound uniform in \(N\), and none is established here.

3. The Effective PDE as a Fitted Closure

The effective PDE is a phenomenological closure for the observables of Section 1.1. Its terms are chosen to mirror the four classes of triadic contribution to the angular-resolved shell energy (radial transfer, angular redistribution within a shell, stretching coupling \(E\) and \(\Omega\), and the exact viscous term), but its coefficients are fitted to data, not derived from the equations, and no theorem controls the error between the closure and the truncated Navier–Stokes system.

3.1 Observables

For a divergence-free velocity field on \(\mathbb{T}^3\), the angular-resolved shell energy at wavenumber \(k\) and polar alignment \(\mu\) is

and \(\Omega_K = \frac{1}{2}\sum_{|\mathbf{k}|=K} |\mathbf{k}|^2 |\hat{u}_{\mathbf{k}}|^2\) is tracked separately.

3.2 The ansatz

where \(\bar{E}(k,t)\) is the isotropic average, \(\Pi(E) = \sqrt{E_k \cdot E_{k+1}}\) is the Leith flux, and \(S(E,\Omega) = k\sqrt{E_K \cdot \Omega_K} \cdot E/E_K\). The viscous term \(-\nu k^2 E\) is hard-coded and not fitted.

In the coupled \((E, \Omega)\) formulation the \(\Omega_K\) equation uses three additional parameters:

Nine coefficients are fitted in the coupled model.

3.3 Status of the closure

Three things determine what the fitted coefficients can and cannot show. First, several 3D fits return coefficients at the bounds of the optimiser's box (\(d_1 = d_2 = 2.000\) in Section 4.5), so those values are constraints, not measurements. Second, the stretching coefficient \(c_5\) varies by two orders of magnitude between \(N = 4\) and \(N = 12\) (Section 4.6), so no converged value is available. Third, the closure is fitted on a fixed window to a finite truncation; a bound on the closure's coefficients is not a bound on the enstrophy of the truncated system, let alone of the equations.

4. Numerical Results

4.1 Direct numerical simulations

We solve the Navier–Stokes equations (or Burgers' equation in 1D) pseudospectrally on \(\mathbb{T}^d\) for \(d = 1, 2, 3\), using fourth-order Runge–Kutta time stepping with \(2/3\)-rule dealiasing.

DNS resolution table:

Four initial conditions are tested per dimension:

• A: Single-shell pulse (quasi-inviscid). Tests inviscid cascade dynamics.
• B: Single-shell pulse (viscous). Tests cross-validation of parameters.
• C: Gaussian packet. Tests wave propagation through multiple shells.
• D: Broad spectrum. Tests generic (non-localised) conditions.

4.2 Fitting procedure

We minimise the forward-integration error:

where \(E^{\mathrm{sim}}\) is obtained by integrating the effective PDE from the DNS initial condition. The enstrophy \(\Omega_K\) is tracked as an independent state variable (breaking the \(\Omega = K^2 E\) degeneracy). Optimisation uses L-BFGS-B with six multi-start seeds.

4.3 Results: 2D Navier–Stokes

The 2D control produces a net stretching of \(\sigma_{\mathrm{net}} = c_5 + c_7 = 0.000\): the absence of vortex stretching in 2D is recovered without being imposed.

4.4 Results: 3D Navier–Stokes

4.5 Fitted coefficients

Fitted coefficients table (coupled \(E\)–\(\Omega\) model, 9 parameters):

Reading the table: in 2D the stretching coefficients \(c_5\) and \(d_3\) are both near zero, as they should be. In 3D at \(N = 8\) the fit returns \(c_5 = -0.518\), \(d_3 = +0.013\) and \(c_2 = +0.128\), and \(d_1 = d_2 = 2.000\) sit at the optimiser's upper bound. These are the values that minimise the forward-integration error for this dataset and window; the 3D fit quality (22–89% relative error, Section 4.4) and the \(N\)-dependence below bound what can be read from them.

4.6 Variation of the fitted coefficients with \(N\)

Convergence of effective coefficients with \(N\):

Across the four truncations the fitted \(c_5\) runs from \(+0.160\) to \(-0.819\), and \(d_3\) from \(-0.249\) to \(+1.823\): the stretching coefficients vary by two orders of magnitude and do not converge with \(N\). The sign of \(c_5\) is positive at \(N = 4\) and negative at \(N = 8, 10, 12\) in each of the 16 per-experiment fits, which coincides with the shell-mixing graph becoming complete for \(K \le 3\) (Lattice fact 1). That coincidence is a property of the fits on these data; because the closure is not derived and its error is not controlled, it is not a statement about the stretching term of the equations.

5. Exact Solutions and Scope

Two exact computations delimit what any argument built on intra-shell relaxation or shell anisotropy can assume.

5.1 The shear flow \(u = A\sin(3x)\,e_y\)

This is an exact solution of the equations on \(\mathbb{T}^3\) (the nonlinearity vanishes identically, and it decays as \(e^{-9\nu t}\)). Its energy sits in two modes of shell \(K = 3\). For it, the intra-shell relaxation rate vanishes identically for all time, and \(\sqrt{E_K A_K}\) grows linearly in the amplitude \(A\) without bound. Any lemma asserting a time-averaged relaxation rate \(\langle\Gamma_K\rangle \ge c_0 K^2\sqrt{\langle E_K\rangle}\) with a universal \(c_0 > 0\), or an anisotropy bound \(\sqrt{E_K}A_K^{1/2} \le \text{const}\), is false for this solution. Constants that were universal would also be inconsistent with the scaling symmetry \(u \mapsto \lambda u(\lambda x, \lambda^2 t)\) of the equations.

5.2 Taylor–Green data

For the Taylor–Green datum, the enstrophy of the truncated system grows before it decays: by a factor \(1.7\) at \(\nu = 0.01\) and \(3.5\) at \(\nu = 0.005\). A cumulative stretching-to-dissipation balance that holds for broad-spectrum data on one \(N = 8\) run therefore does not hold for general smooth data, and monotone decrease of \(\Omega\) is not a property of the equations.

6. Discussion

6.1 The role of the 2D control

Any closure meant to describe 3D must first be tested against 2D, where the answer is known. The 2D fit reproduces the Kraichnan dual cascade (inverse energy, forward enstrophy), returns \(\sigma_{\mathrm{net}} = 0\) without imposing it, and achieves sub-3% accuracy on all four initial conditions.

6.2 Limitations

3D fit quality. The 3D universal fit achieves 22–23% relative error on pulse initial conditions and 72–89% on packet and broad-spectrum initial conditions. This is enough to read the sign and order of magnitude of the fitted coefficients on a given dataset, not to make quantitative claims.

Resolution dependence. The 3D simulations span \(N = 4\) through \(N = 12\), all far from resolving a dissipation range. The fitted stretching coefficients do not converge over this range.

Dimensional consistency. The fitted coefficients carry the units of the window and amplitude they were fitted on; a closure with fixed numerical coefficients cannot be invariant under the scaling symmetry of the equations, which is a further reason its coefficients cannot enter a bound.

6.3 Comparison with Kolmogorov scaling

The fitted 3D Leith flux coefficient \(c_4 = +0.220\) has the sign of a forward energy cascade, and \(c_4 < 0\) in 2D has the sign of the inverse cascade, consistent with K41 and Kraichnan respectively. This is a check that the flux term of the closure is doing what it was designed to do.

7. Conclusion

The angular-resolved shell energy \(E(K,\mu,t)\) is a useful observable for the Galerkin truncations of 3D Navier–Stokes: it separates transfer between shells from redistribution within a shell, and the two lattice facts (complete shell mixing graph for \(N \ge 2K+1\); stretching expressible through a shell anisotropy) describe how the nonlinearity acts on it. The effective PDE fitted to that observable reproduces the 2D data within 2% with a vanishing stretching coefficient, and in 3D returns a stretching coefficient whose sign changes across the saturation threshold in each of the 16 per-experiment fits.

The 3D closure is a fitted nine-coefficient description of the data. Its fits sit at the optimiser's bounds, its stretching coefficient varies by two orders of magnitude between \(N = 4\) and \(N = 12\), it is not derived from the equations, and no theorem controls its error. The exact shear flow \(u = A\sin(3x)\,e_y\) shows that intra-shell relaxation can vanish identically while \(\sqrt{E_K A_K}\) is unbounded, so no relaxation lemma or anisotropy bound with universal constants is available. No claim of global regularity is made. What remains is the classical conditional statement of Section 2, the two lattice facts, and the effective-PDE fits as a description of the data.

References

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Citation

@article{higgins2026angular,
  title={Angular Relaxation on the Integer Lattice and Global
         Regularity of 3D Navier--Stokes},
  author={Higgins, Rod},
  year={2026},
  doi={10.5281/zenodo.22710240},
  note={v2},
  url={https://lab.senuamedia.com/papers/effective-pde-ns-regularity.html}
}