| _version_ | 1866901536408338432 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | The three-dimensional incompressible Navier-Stokes equations are fundamental to fluid dynamics, yet their global regularity for smooth initial data remains one of the most significant unsolved problems in mathematics, a Millennium Prize Problem. This paper proposes a novel characterization of the regularity conjecture through the lens of geometric invariants. We investigate how the behavior and boundedness of specific geometric quantities, such as helicity, enstrophy, and the curvature of streamlines, could dictate the global existence and smoothness of solutions. By exploring the interplay between the topological and geometric features of fluid flow and the analytical properties of the Navier-Stokes equations, this work aims to establish a theoretical framework where regularity is tied to the geometric coherence and evolution of vortex structures. We hypothesize that if certain geometric invariants remain bounded or satisfy specific growth conditions, then the solutions must be regular. This approach offers a new perspective beyond traditional energy estimates and scaling arguments, suggesting that a deeper understanding of fluid geometry could unlock the secrets to global regularity. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17799237 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Regularity Conjecture for 3D Navier-Stokes: A New Characterization through Geometric Invariants Revista, Zen MATH, 10 The three-dimensional incompressible Navier-Stokes equations are fundamental to fluid dynamics, yet their global regularity for smooth initial data remains one of the most significant unsolved problems in mathematics, a Millennium Prize Problem. This paper proposes a novel characterization of the regularity conjecture through the lens of geometric invariants. We investigate how the behavior and boundedness of specific geometric quantities, such as helicity, enstrophy, and the curvature of streamlines, could dictate the global existence and smoothness of solutions. By exploring the interplay between the topological and geometric features of fluid flow and the analytical properties of the Navier-Stokes equations, this work aims to establish a theoretical framework where regularity is tied to the geometric coherence and evolution of vortex structures. We hypothesize that if certain geometric invariants remain bounded or satisfy specific growth conditions, then the solutions must be regular. This approach offers a new perspective beyond traditional energy estimates and scaling arguments, suggesting that a deeper understanding of fluid geometry could unlock the secrets to global regularity. |
| title | The Regularity Conjecture for 3D Navier-Stokes: A New Characterization through Geometric Invariants |
| url | https://doi.org/10.5281/zenodo.17799237 |