A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation
| dc.creator | Tao, Terence | |
| dc.date | 2007-10-08 | |
| dc.date | 2009-05-21 | |
| dc.date.accessioned | 2026-07-07T13:16:36Z | |
| dc.date.available | 2026-07-07T13:16:36Z | |
| dc.description | The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\R/\Z)^3 \to \R^3$ there exists a global smooth solution u. In this note we observe (using a simple compactness argument) that this qualitative question is equivalent to the more quantitative assertion that there exists a non-decreasing function $F: \R^+ \to \R^+$ for which one has a local-in-time \emph{a priori} bound $$ \| u(T) \|_{H^1_x((\R/\Z)^3)} \leq F(\|u_0\|_{H^1_x((\R/\Z)^3)})$$ for all $0 < T \leq 1$ and all smooth solutions $u: [0,T] \times (\R/\Z)^3 \to \R^3$ to the Navier-Stokes system. We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound. | |
| dc.description | 12 pages, no figures. More minor corrections (not appearing in the published version) | |
| dc.identifier | https://arxiv.org/abs/0710.1604 | |
| dc.identifier | http://arxiv.org/abs/0710.1604 | |
| dc.identifier | Dynamics of PDE 4 (2007), 293--302 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230844 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35Q30 | |
| dc.title | A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation | |
| dc.type | text |