An Effective PDE for Shell-Angular Energy and Global Regularity of 3D Navier–Stokes
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2026
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| _version_ | 1866902035991887872 |
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| 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 and absorbed into the viscous term via a bootstrap argument.</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.</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 with the stretching coefficients vanishing automatically (c<sub>5</sub> ≈ 0, d<sub>3</sub> ≈ 0), confirming the framework against a known-regular case. The 3D fit gives a dissipative stretching coefficient (c<sub>5</sub> < 0 at both N=8 and N=10), confirmed by a convergence study across truncations N = 4, 8, 10. The stretching coefficient 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), and remains negative thereafter.</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 uniform enstrophy 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_19553710 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| 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 and absorbed into the viscous term via a bootstrap argument.</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.</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 with the stretching coefficients vanishing automatically (c<sub>5</sub> ≈ 0, d<sub>3</sub> ≈ 0), confirming the framework against a known-regular case. The 3D fit gives a dissipative stretching coefficient (c<sub>5</sub> < 0 at both N=8 and N=10), confirmed by a convergence study across truncations N = 4, 8, 10. The stretching coefficient 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), and remains negative thereafter.</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 uniform enstrophy 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.19553710 |