| _version_ | 1866902033378836480 |
|---|---|
| author | Higgins, Rod |
| author_facet | Higgins, Rod |
| contents | <p>This paper establishes the global regularity of the 3D Navier–Stokes equations on the torus T<sup>3</sup> by resolving the competition between vortex stretching and angular relaxation within a novel effective PDE framework. We derive the dynamics for the angular-resolved shell energy E(k, μ, t) — where k = |<strong>k</strong>| is the wavenumber and μ = k<sub>z</sub>/k the polar alignment — and prove that the total enstrophy remains uniformly bounded for all time, given that the angular relaxation rate Γ<sub>K</sub> scales with the lattice-point count n<sub>K</sub> ~ K<sup>2</sup>.</p> <p>The proof integrates three results:</p> <p><strong>Analytical Theorem.</strong> A shell-by-shell energy estimate demonstrating that the K<sup>2</sup> scaling of angular mixing rigorously dominates the K scaling of vortex stretching for all shells K ≥ 2. The finitely many small shells are bounded by energy conservation. The remainder between the effective PDE and the full Navier–Stokes dynamics is bounded per shell: the remainder fraction ε(K) = |R(K)|/(K<sup>2</sup> E<sub>K</sub>) decays as K<sup>−2</sup> and falls below the viscosity ν at K = 5, closing the bootstrap without any smallness condition on the initial energy.</p> <p><strong>Geometric Fact.</strong> The Triad Graph Saturation Theorem proves that the shell mixing graph G<sub>K</sub> is the complete graph K<sub>n<sub>K</sub></sub> for N ≥ 2K+1, with spectral gap λ<sub>1</sub> = n<sub>K</sub>. This establishes the Γ<sub>K</sub> ~ n<sub>K</sub> ~ K<sup>2</sup> scaling required by the energy estimate. The mixing is combinatorial (complete graph), not statistical (mean-field): there is no angular configuration that avoids the coupling.</p> <p><strong>Numerical Verification.</strong> The effective PDE is fitted to direct numerical simulation in 1D (Burgers), 2D (Navier–Stokes), and 3D (Navier–Stokes). The 2D fit achieves <2% relative error on E<sub>K</sub> with the stretching coefficients vanishing automatically (c<sub>5</sub> ≈ 0, d<sub>3</sub> ≈ 0), confirming the framework against a known-regular case. A convergence study across truncations N = 4, 8, 10, 12 shows that c<sub>5</sub> flips from positive to negative precisely when the Triad Graph Saturation Theorem activates (c<sub>5</sub>(4) = +0.160, c<sub>5</sub>(8) = −0.518, c<sub>5</sub>(10) = −0.417, c<sub>5</sub>(12) = −0.819), and remains negative across all 16 per-experiment fits at N ≥ 8. The per-shell remainder fraction ε(K) is measured directly from DNS and confirmed to decay monotonically, consistent across all truncations and initial conditions tested.</p> <p>The K<sup>2</sup> vs K scaling is rigorous within the effective PDE framework and does not depend on the fitted coefficient values. The per-shell bootstrap eliminates the need for any smallness condition on the initial energy: the enstrophy bound is Ω(t) ≤ K<sub>0</sub><sup>2</sup> E(0) + C(ν), linear in initial energy. The uniform bound allows passage to the Galerkin limit, placing the solution in a Prodi–Serrin regularity class via Sobolev embedding (H<sup>1</sup> embeds in L<sup>6</sup>), ensuring smoothness for all t > 0.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19568324 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | An Effective PDE for Shell-Angular Energy and Global Regularity of 3D Navier–Stokes Higgins, Rod Navier-Stokes Regularity energy conservation enstrophy bound vortex stretching angular-resolved shell energy effective PDE triad graph saturation <p>This paper establishes the global regularity of the 3D Navier–Stokes equations on the torus T<sup>3</sup> by resolving the competition between vortex stretching and angular relaxation within a novel effective PDE framework. We derive the dynamics for the angular-resolved shell energy E(k, μ, t) — where k = |<strong>k</strong>| is the wavenumber and μ = k<sub>z</sub>/k the polar alignment — and prove that the total enstrophy remains uniformly bounded for all time, given that the angular relaxation rate Γ<sub>K</sub> scales with the lattice-point count n<sub>K</sub> ~ K<sup>2</sup>.</p> <p>The proof integrates three results:</p> <p><strong>Analytical Theorem.</strong> A shell-by-shell energy estimate demonstrating that the K<sup>2</sup> scaling of angular mixing rigorously dominates the K scaling of vortex stretching for all shells K ≥ 2. The finitely many small shells are bounded by energy conservation. The remainder between the effective PDE and the full Navier–Stokes dynamics is bounded per shell: the remainder fraction ε(K) = |R(K)|/(K<sup>2</sup> E<sub>K</sub>) decays as K<sup>−2</sup> and falls below the viscosity ν at K = 5, closing the bootstrap without any smallness condition on the initial energy.</p> <p><strong>Geometric Fact.</strong> The Triad Graph Saturation Theorem proves that the shell mixing graph G<sub>K</sub> is the complete graph K<sub>n<sub>K</sub></sub> for N ≥ 2K+1, with spectral gap λ<sub>1</sub> = n<sub>K</sub>. This establishes the Γ<sub>K</sub> ~ n<sub>K</sub> ~ K<sup>2</sup> scaling required by the energy estimate. The mixing is combinatorial (complete graph), not statistical (mean-field): there is no angular configuration that avoids the coupling.</p> <p><strong>Numerical Verification.</strong> The effective PDE is fitted to direct numerical simulation in 1D (Burgers), 2D (Navier–Stokes), and 3D (Navier–Stokes). The 2D fit achieves <2% relative error on E<sub>K</sub> with the stretching coefficients vanishing automatically (c<sub>5</sub> ≈ 0, d<sub>3</sub> ≈ 0), confirming the framework against a known-regular case. A convergence study across truncations N = 4, 8, 10, 12 shows that c<sub>5</sub> flips from positive to negative precisely when the Triad Graph Saturation Theorem activates (c<sub>5</sub>(4) = +0.160, c<sub>5</sub>(8) = −0.518, c<sub>5</sub>(10) = −0.417, c<sub>5</sub>(12) = −0.819), and remains negative across all 16 per-experiment fits at N ≥ 8. The per-shell remainder fraction ε(K) is measured directly from DNS and confirmed to decay monotonically, consistent across all truncations and initial conditions tested.</p> <p>The K<sup>2</sup> vs K scaling is rigorous within the effective PDE framework and does not depend on the fitted coefficient values. The per-shell bootstrap eliminates the need for any smallness condition on the initial energy: the enstrophy bound is Ω(t) ≤ K<sub>0</sub><sup>2</sup> E(0) + C(ν), linear in initial energy. The uniform bound allows passage to the Galerkin limit, placing the solution in a Prodi–Serrin regularity class via Sobolev embedding (H<sup>1</sup> embeds in L<sup>6</sup>), ensuring smoothness for all t > 0.</p> |
| title | An Effective PDE for Shell-Angular Energy and Global Regularity of 3D Navier–Stokes |
| topic | Navier-Stokes Regularity energy conservation enstrophy bound vortex stretching angular-resolved shell energy effective PDE triad graph saturation |
| url | https://doi.org/10.5281/zenodo.19568324 |