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M. Bauer, B. Kolev, S. C. Preston:
"Geometric investigations of a vorticity model equation";
Journal of Differential Equations, 260 (2016), 1; S. 478 - 516.

This article consists of a detailed geometric study of the one-dimensional vorticity model equation
View the MathML sourceωt+uωx+2ωux=0,ω=Hux,t∈R,x∈S1,

which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on Diff(S1)Diff(S1) when the latter is endowed with the right-invariant homogeneous View the MathML sourceH˙1/2-metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-Córdoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to Córdoba-Córdoba.

Generalized CLM equation; Fredholmness of the Riemannian exponential mapping; Sobolev metrics of fractional order

http://dx.doi.org/10.1016/j.jde.2015.09.030