Remarks on the Minimizing Geodesic Problem in Inviscid Incompressible Fluid Mechanics

TL;DR

This paper explores the properties of $L^2$ minimizing geodesics in the group of volume-preserving maps, relating to ideal fluid motion, and discusses their uniqueness and regularity limitations in infinite-dimensional settings.

Contribution

It demonstrates that the uniqueness of these geodesics is an infinite-dimensional phenomenon and clarifies the inherent limitations of their partial regularity.

Findings

Uniqueness is specific to infinite-dimensional groups like $SDiff(D)$.

Partial regularity of pressure fields is necessarily limited.

Finite-dimensional analogs like O(3) do not exhibit the same uniqueness properties.

Abstract

We consider $L^2$ minimizing geodesics along the group of volume preserving maps $SDiff(D)$ of a given 3-dimensional domain $D$. The corresponding curves describe the motion of an ideal incompressible fluid inside $D$ and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: 1) the uniqueness property can be viewed as an infinite dimensional phenomenon (related to the possibility of relaxing the corresponding minimization problem by convex optimization), which is false for finite dimensional configuration spaces such as O(3) for the motion of rigid bodies; 2) the unconditional partial regularity is necessarily limited.