Global Regularity of the 3D Incompressible Navier–Stokes Equations via Energy Estimates, Bootstrap Closure, and Vorticity Control
Fuente:
Zenodo
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Recurso digital |
| Sprache: | Englisch |
| Veröffentlicht: |
Zenodo
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866902128959684608 |
|---|---|
| author | Kevin Fathi |
| author_facet | Kevin Fathi |
| contents | <p>We prove the global existence and smoothness of solutions to the three-dimensional incompressible Navier–Stokes equations on R3 for smooth, divergence-free initial data with finite energy. The proof proceeds by deriving precise a priori bounds on the vorticity and higher Sobolev norms, closing a bootstrap argument through classical energy estimates and functional inequalities. A central element of the argument is the control of the nonlinear vortex stretching term and suppression of high-frequency energy accumulation through viscous damping. We apply the Beale–Kato–Majda continuation criterion, showing that the L∞ norm of vorticity remains integrable in time due to uniform H^3 bounds. As a result, the solution persists globally<br>and remains smooth for all t > 0. The proof is entirely classical, relying on deterministic partial differential equation techniques without probabilistic or numerical assumptions.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15384518 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Global Regularity of the 3D Incompressible Navier–Stokes Equations via Energy Estimates, Bootstrap Closure, and Vorticity Control Kevin Fathi Navier-Stokes equations Millennium Prize Problem global regularity fluid dynamics partial differential equations Sobolev spaces energy inequality Beale–Kato–Majda criterion vorticity mathematical analysis functional analysis Leray solutions weak solutions H1 estimates smooth solutions global existence PDE blow-up analytic proof incompressible flow mathematics preprint <p>We prove the global existence and smoothness of solutions to the three-dimensional incompressible Navier–Stokes equations on R3 for smooth, divergence-free initial data with finite energy. The proof proceeds by deriving precise a priori bounds on the vorticity and higher Sobolev norms, closing a bootstrap argument through classical energy estimates and functional inequalities. A central element of the argument is the control of the nonlinear vortex stretching term and suppression of high-frequency energy accumulation through viscous damping. We apply the Beale–Kato–Majda continuation criterion, showing that the L∞ norm of vorticity remains integrable in time due to uniform H^3 bounds. As a result, the solution persists globally<br>and remains smooth for all t > 0. The proof is entirely classical, relying on deterministic partial differential equation techniques without probabilistic or numerical assumptions.</p> |
| title | Global Regularity of the 3D Incompressible Navier–Stokes Equations via Energy Estimates, Bootstrap Closure, and Vorticity Control |
| topic | Navier-Stokes equations Millennium Prize Problem global regularity fluid dynamics partial differential equations Sobolev spaces energy inequality Beale–Kato–Majda criterion vorticity mathematical analysis functional analysis Leray solutions weak solutions H1 estimates smooth solutions global existence PDE blow-up analytic proof incompressible flow mathematics preprint |
| url | https://doi.org/10.5281/zenodo.15384518 |