Dual pairs in fluid dynamics

TL;DR

This paper rigorously explores the dual pair structures in ideal fluid dynamics and the Camassa-Holm equation, revealing connections to central extensions of diffeomorphism groups and providing detailed proofs of transitivity properties.

Contribution

It introduces a precise framework for dual pair structures in fluid dynamics and extends the understanding of their relation to central extensions of diffeomorphism groups.

Findings

Dual pair structures are rigorously characterized in ideal fluid and Camassa-Holm equations.

Central extensions of diffeomorphism groups naturally arise from momentum map definitions.

Transitivity results for the dual pair structures are established with detailed proofs.

Abstract

This paper is a rigorous study of the dual pair structure of the ideal fluid and the dual pair structure for the $n$-dimensional Camassa-Holm (EPDiff) equation, including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads naturally to central extensions of diffeomorphism groups such as the group of quantomorphisms and the Ismagilov central extension.