Geometric investigations of a vorticity model equation

This article consists of a detailed geometric study of the one-dimensional vorticity model equationωt+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 H˙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.