# Vortex sheets and diffeomorphism groupoids

## TL;DR

This paper extends Arnold's geometric approach to ideal hydrodynamics to include vortex sheets by introducing a groupoid framework of volume-preserving diffeomorphisms with common interfaces, providing new geometric and Hamiltonian insights.

## Contribution

It develops a novel groupoid-based framework for modeling vortex sheet dynamics within the geometric and Hamiltonian structures of fluid flows.

## Key findings

- Vortex sheet dynamics are modeled using a groupoid of diffeomorphisms with common interfaces.
- A general framework for Euler-Arnold equations on groupoids with invariant metrics is established.
- Connections between vortex sheets and groupoid structures are elucidated.

## Abstract

In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler-Arnold equations for geodesics on groupoids equipped with one-sided invariant metrics.