Least action principles for incompressible flows and geodesics between shapes

TL;DR

This paper explores the geometric and variational principles underlying incompressible fluid flows and shape evolution, revealing connections to Wasserstein distances, Euler sprays, and two-fluid mixture models.

Contribution

It introduces a least-action framework for shape flows using geodesic equations, linking incompressible Euler equations to optimal transport and shape analysis.

Findings

Any two shapes of equal volume can be approximately connected by Euler sprays.

The infimum of the action equals the Wasserstein distance squared, rarely attained in higher dimensions.

Wasserstein geodesics solve a pressureless Euler system as weak limits of Euler sprays.

Abstract

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: Any two shapes of equal volume can be approximately connected by an Euler spray---a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays. Each such Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a two-fluid mixture theory corresponding to incompressible fluid mixed with vacuum.