Niveau: Supérieur, Doctorat, Bac+8
An extension of the Beale-Kato-Majda criterion for the Euler equations Fabrice Planchon ? Abstract The Beale-Kato-Majda criterion asserts that smooth solutions to the Euler equations remain bounded past T as long as ∫ T 0 ???∞dt is finite, ? being the vorticity. We show how to replace this by a weaker statement, on supj ∫ T 0 ?∆j??∞dt, where ∆j is a frequency localization around |?| ≈ 2j . Introduction The incompressible Euler equations read ? ?? ?? ∂u ∂t + u · ?u = ??p, ? · u = 0, u(x, 0) = u0(x), x ? Rn, t ≥ 0. (1) These equations are known to be locally well-posed for data u0 ? Hs, s > n 2 +1, or more generallyW sp with s? np > 1 (see [6] and references therein). In a celebrated paper, Beale-Kato-Majda gave the following criterion for blow- up: if blow-up occurs at time T , then necessarily, ∫ T 0 ???∞dt = +∞,(2) where ? = ?? u is the vorticity. One may rephrase it as: if ∫ T 0 ???∞dt < +∞,(3) ?Laboratoire Analyse, Geometrie & Applications, UMR 7539, Institut Galilee, Univer- site Paris 13, 99 avenue J.
- criterion asserts
- can control ∫
- solution can
- beale-kato-majda criterion
- past time
- remain bounded past