The Polozov Law and the Regularity of 3D Navier-Stokes
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2025
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| author | Polozov, Andrei |
| author_facet | Polozov, Andrei |
| contents | <p>This paper introduces the Polozov Law, a groundbreaking nonlinear dissipation mechanism that ensures global regularity for the three-dimensional Navier-Stokes equations. By modifying the vorticity evolution equation with a superlinear damping term, we rigorously prove that under the condition <strong>p > 2</strong>, the vorticity remains globally bounded, preventing finite-time singularities.</p> <p>A key aspect of this work is demonstrating that the proposed dissipation term <strong>arises intrinsically</strong> from the Navier-Stokes equations without requiring external modifications. Using a combination of <strong>functional analysis, renormalization techniques, and spectral energy transfer analysis</strong>, we establish that this nonlinear damping effect is an emergent property of the viscosity term at small scales.</p> <p>Furthermore, we prove that the global control of vorticity ensures the <strong>smoothness of velocity fields</strong> in all Sobolev spaces, confirming the absence of blow-up scenarios. The results are validated through rigorous energy estimates, attractor theory, and asymptotic stability analysis. This proof conclusively addresses the <strong>Millennium Prize Problem on Navier-Stokes regularity</strong>, as posed by the <strong>Clay Mathematics Institute</strong>.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_14952786 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Polozov Law and the Regularity of 3D Navier-Stokes Polozov, Andrei <p>This paper introduces the Polozov Law, a groundbreaking nonlinear dissipation mechanism that ensures global regularity for the three-dimensional Navier-Stokes equations. By modifying the vorticity evolution equation with a superlinear damping term, we rigorously prove that under the condition <strong>p > 2</strong>, the vorticity remains globally bounded, preventing finite-time singularities.</p> <p>A key aspect of this work is demonstrating that the proposed dissipation term <strong>arises intrinsically</strong> from the Navier-Stokes equations without requiring external modifications. Using a combination of <strong>functional analysis, renormalization techniques, and spectral energy transfer analysis</strong>, we establish that this nonlinear damping effect is an emergent property of the viscosity term at small scales.</p> <p>Furthermore, we prove that the global control of vorticity ensures the <strong>smoothness of velocity fields</strong> in all Sobolev spaces, confirming the absence of blow-up scenarios. The results are validated through rigorous energy estimates, attractor theory, and asymptotic stability analysis. This proof conclusively addresses the <strong>Millennium Prize Problem on Navier-Stokes regularity</strong>, as posed by the <strong>Clay Mathematics Institute</strong>.</p> |
| title | The Polozov Law and the Regularity of 3D Navier-Stokes |
| url | https://doi.org/10.5281/zenodo.14952786 |