Geometric investigations of a vorticity model equation

Abstract

This article consists of a detailed geometric study of the one-dimensional vorticity model equation $$\ddot{I}{‰}_t+u\ddot{I}{‰}_x+2\ddot{I}‰u_x=0,\ddot{I}‰=H{u}_x,t\in \mathbb{R},x\in S^1,$$ which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on $\mathrm{Diff}(S^1)$ when the latter is endowed with the right-invariant homogeneous ${\dot{H}}^{1/2}$-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.

30 pages; added references; corrected typos