Conjugate point criteria on the area-preserving diffeomorphism group

TL;DR

This paper introduces a new criterion for detecting conjugate points in steady 2D Euler flows, improving previous methods and applicable to rotational cells, with several illustrative examples and applications to Kolmogorov flows.

Contribution

It presents a novel sufficient criterion for conjugate points in steady Euler flows, applicable in rotational cells, and offers a general construction method for steady flows with specific properties.

Findings

New criterion improves detection of conjugate points

Criterion applies to all known conjugate points in rotational symmetry

Examples include conjugate points in Kolmogorov flows

Abstract

This paper answers some questions about conjugate points along the geodesics corresponding to steady 2D Euler flows, posed by a paper of Drivas-Misiolek-Shi-Yoneda. We present a new sufficient criterion for the existence of conjugate points, which improves on the criterion of Misiolek. It applies in any rotational cell of a steady 2D Euler flow, and in case of rotational symmetry it captures all known conjugate points. We give a general construction of the surfaces that admit a steady fluid with given area form, velocity profile, and vorticity profile, and from this we show how to detect conjugate points in a single rotational cell of a steady flow. When the velocity profile has a local extremum, the criterion becomes particularly simple. Several examples are provided, and in an appendix we use the Misio{\l}ek criterion to give some new examples of conjugate points along Kolmogorov flows on the torus.