Geometric investigations of a vorticity model equation

TL;DR

This paper provides a geometric analysis of a vorticity model equation, revealing properties like non-Fredholm exponential map, unbounded curvature, and conditions for blow-up, extending previous results to broader contexts.

Contribution

It proves the non-Fredholmness of the exponential map, analyzes the curvature, and extends blow-up criteria for the vorticity model equation.

Findings

Exponential map of the right-invariant metric is not Fredholm.

Sectional curvature is locally unbounded.

A generalized blow-up criterion is established for wider initial conditions.

Abstract

This article consists of a detailed geometric study of the one-dimensional vorticity model equation $$\omega_{t} + u\omega_{x} + 2\omega u_{x} = 0, \qquad \omega = H u_{x}, \qquad t\in\mathbb{R},\; x\in S^{1}\,,$$ which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on $\operatorname{Diff}(S^{1})$ 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-C\'ordoba 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 C\'ordoba-C\'ordoba.