Global Regularity of the Three-Dimensional Navier–Stokes Equations via Spectral Gap and Frequency Rigidity
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| Langue: | anglais |
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2025
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| _version_ | 1866901438913839104 |
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| author | Rodrigues, Vinicius Ramos |
| author_facet | Rodrigues, Vinicius Ramos |
| contents | <p>We prove unconditional global regularity for the incompressible three-dimensional Navier–Stokes equations on the periodic torus $\mathbb{T}^3_L$, resolving the Clay Mathematics Institute Millennium Prize Problem in this setting. The proof establishes an intrinsic spectral gap $\lambda_1(t) \ge \varepsilon_0 > 0$ through three fundamental lemmas: (i) a quantitative Cheeger inequality on Whitney–decomposed graphs yielding $\lambda^{\mathrm{disc}}_1 \ge c_2\, h/(4\ell^2)$ with explicit $c_h = 1/4$; (ii) Mosco $\Gamma$–convergence transporting discrete coercivity to the continuum with error $O(\sqrt{\delta})$; (iii) thickness non-collapse via De Giorgi iteration establishing $\delta(t) \ge \delta_* > 0$ uniformly. These combine to prove that the spectral gap $\gamma_* \ge 1/512$ is always satisfied, and not assumed. </p> <p>The conditions CVA–HF (H1)–(H2), controlling high-frequency vorticity behavior via Littlewood–Paley decomposition with an explicit error term $M(t)$ satisfying $\int_0^{\infty} M(t)\,dt < \infty$, are shown to be automatic consequences of the REFP framework, rather than external hypotheses. Coupled with the Beale–Kato–Majda criterion, this yields</p> <p>\[</p> <p>\int_0^T \|\omega\|_{L^\infty}\,dt < \infty \quad \text{for all } T > 0,</p> <p>\]</p> <p>establishing global smoothness for all divergence-free $u_0 \in L^2$ with finite energy. Two independent proof pathways are provided: a constructive route via thickness preservation and a contradictory route via violation of the local energy inequality. All constants are explicit and depend only on $(\nu, L, \|u_0\|_{L^2})$. The result is unconditional: no external hypotheses beyond the Navier–Stokes equations and the energy inequality are required.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17764395 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Global Regularity of the Three-Dimensional Navier–Stokes Equations via Spectral Gap and Frequency Rigidity Rodrigues, Vinicius Ramos Navier–Stokes equations; Global regularity; Spectral gap; Vorticity dynamics; Millennium Prize Problem; REFP framework; Beale–Kato–Majda criterion; LittlewoodPaley theory <p>We prove unconditional global regularity for the incompressible three-dimensional Navier–Stokes equations on the periodic torus $\mathbb{T}^3_L$, resolving the Clay Mathematics Institute Millennium Prize Problem in this setting. The proof establishes an intrinsic spectral gap $\lambda_1(t) \ge \varepsilon_0 > 0$ through three fundamental lemmas: (i) a quantitative Cheeger inequality on Whitney–decomposed graphs yielding $\lambda^{\mathrm{disc}}_1 \ge c_2\, h/(4\ell^2)$ with explicit $c_h = 1/4$; (ii) Mosco $\Gamma$–convergence transporting discrete coercivity to the continuum with error $O(\sqrt{\delta})$; (iii) thickness non-collapse via De Giorgi iteration establishing $\delta(t) \ge \delta_* > 0$ uniformly. These combine to prove that the spectral gap $\gamma_* \ge 1/512$ is always satisfied, and not assumed. </p> <p>The conditions CVA–HF (H1)–(H2), controlling high-frequency vorticity behavior via Littlewood–Paley decomposition with an explicit error term $M(t)$ satisfying $\int_0^{\infty} M(t)\,dt < \infty$, are shown to be automatic consequences of the REFP framework, rather than external hypotheses. Coupled with the Beale–Kato–Majda criterion, this yields</p> <p>\[</p> <p>\int_0^T \|\omega\|_{L^\infty}\,dt < \infty \quad \text{for all } T > 0,</p> <p>\]</p> <p>establishing global smoothness for all divergence-free $u_0 \in L^2$ with finite energy. Two independent proof pathways are provided: a constructive route via thickness preservation and a contradictory route via violation of the local energy inequality. All constants are explicit and depend only on $(\nu, L, \|u_0\|_{L^2})$. The result is unconditional: no external hypotheses beyond the Navier–Stokes equations and the energy inequality are required.</p> |
| title | Global Regularity of the Three-Dimensional Navier–Stokes Equations via Spectral Gap and Frequency Rigidity |
| topic | Navier–Stokes equations; Global regularity; Spectral gap; Vorticity dynamics; Millennium Prize Problem; REFP framework; Beale–Kato–Majda criterion; LittlewoodPaley theory |
| url | https://doi.org/10.5281/zenodo.17764395 |