Axisymmetric diffeomorphisms and ideal fluids on Riemannian 3-manifolds

TL;DR

This paper explores the Riemannian geometry of 3D axisymmetric ideal fluids, proving properties of the exponential map and defining special diffeomorphism classes with applications to Euler equations on various geometries.

Contribution

It introduces the concept of axisymmetric and swirl-free diffeomorphisms, showing they form a totally geodesic submanifold and analyzing the exponential map in this context.

Findings

The $L^2$ exponential map is Fredholm along small swirl axisymmetric flows.

Axisymmetric and swirl-free diffeomorphisms form a totally geodesic submanifold.

Derived axisymmetric Euler equations on Thurston's eight model geometries.

Abstract

We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston's eight model geometries.