Journal
PHYSICA D-NONLINEAR PHENOMENA
Volume 239, Issue 14, Pages 1225-1231Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.physd.2009.07.009
Keywords
Spectral dynamics; Finite-time breakdown; Restricted Euler equations; Critical threshold; 3D and 4D equations; Global regularity
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We are concerned with the critical threshold phenomena in the restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for the 3D and 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that the 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic. (C) 2009 Elsevier B.V. All rights reserved.
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