On the critical dissipative quasi-geostrophic equation

Citation
P. Constantin et al., On the critical dissipative quasi-geostrophic equation, INDI MATH J, 50, 2001, pp. 97-107
Citations number
14
Categorie Soggetti
Mathematics
Journal title
INDIANA UNIVERSITY MATHEMATICS JOURNAL
ISSN journal
00222518 → ACNP
Volume
50
Year of publication
2001
Pages
97 - 107
Database
ISI
SICI code
0022-2518(200121)50:<97:OTCDQE>2.0.ZU;2-3
Abstract
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model, then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power, the dissipation appears to be ins ufficient. For instance, it is not known if the critical dissipative QG equ ation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of t he critical dissipative QG equation for initial data that have small L-infi nity norm. The importance of an L-infinity smallness condition is due to th e fact that L-infinity is a conserved norm for the non-dissipative QG equat ion and is non-increasing on all solutions of the dissipative QG, irrespect ive of size.