Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

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.

On quasisymmetric plasma equilibria sustained by small force

“…Both of these difficulties can be traced to the fact that ξ need not be a Killing field with respect to the Euclidean metric. To circumvent these issues, inspired by Lichtenfelz, Misiolek & Preston (2019) and , we replace the metric structure of (R 3 , δ) with (R 3 , g) for a metric g for which ξ is a Killing field. The resulting magnetic field will not satisfy the usual MHS equations (1.1), but provided ξ is sufficiently close to Killing for the Euclidean metric, the error will be small.…”

Two-Point Boundary Value Problems on Diffeomorphism Groups

“…Axisymmetric Diffeomorphisms. Let M be a 3-dimensional manifold equipped with a smooth Killing field K. Following [21] we define a divergence-free vector field u on M to be axisymmetric if it commutes with the Killing field: [K, u] = 0. We denote the set of H s axisymmetric vector fields by T e A s µ (M ).…”

“…In our argument for this section, the swirl-free condition will deliver a conservation law which will play the same role that conservation of vorticity did in the previous section. It is shown in [21] that the swirl of an axisymmetric velocity field is transported by its flow. More precisely, if u 0 ∈ T e A s µ and γ(t) is the corresponding geodesic in A s µ then g(u, K)•γ(t) = g(u 0 , K) as long as it is defined.…”

“…We denote the space of swirl-free axisymmetric vector fields by T e A s µ,0 . The proof of the following lemma can be found in [21].…”

“…His method involves constructing a connection on the full diffeomorphism group which turns the volume-preserving diffeomrphisms into a totally geodesic submanifold. In the proof of Theorem 1.2 we make use of the recent work of Lichtenfelz, Misio lek and Preston [21] on the Euler equations in 3 dimensions with axisymmetric swirl-free initial data.…”