Geometric Investigations of a Vorticity Model Equation

This article consists of a detailed geometric study of the one-dimensional vorticity model equation ωt + uωx + 2ωux = 0, ω = Hux, t ∈ R, x ∈ S 1 , which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on Diff(S) when the latter is endowed with the rightinvariant homogeneous Ḣ–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.