Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T00:59:10.738Z Has data issue: false hasContentIssue false

A sufficient condition for a finite-time $L_2 $ singularity of the 3d Euler Equations

Published online by Cambridge University Press:  21 October 2005

XINYU HE
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL. e-mail: [email protected]

Abstract

A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)