Communications in Mathematical Sciences

Volume 9 (2011)

Number 3

The generalized Constantin-Lax-Majda equation revisited

Pages: 929 – 936

(Fast Communication)

DOI: https://dx.doi.org/10.4310/CMS.2011.v9.n3.a12

Author

Marcus Wunsch (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan)

Abstract

We continue our study of the generalized Constantin-Lax-Majda equation, which is an evolution equation in one space dimension modeling three-dimensional vorticity dynamics. First, we show that the BMO-norm of the vorticity controls the singularity formation for smooth solutions if the parameter a equals 2, and we proceed by demonstrating that there are small solutions which exist indefinitely in the presence of viscosity if a≤−2.

Keywords

generalized Constantin-Lax-Majda equation, Beale-Kato-Majda blowup criterion, small solution

2010 Mathematics Subject Classification

35B44, 35Q35, 76B03

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Published 11 March 2011