[Paper Review] Least action principles for incompressible flows and optimal transport between shapes

Jian‐Guo Liu, Robert L. Pego+1 more|arXiv (Cornell University)|Apr 12, 2016

Geometric Analysis and Curvature Flows56 references3 citations

TL;DR

This paper formulates incompressible fluid flows between shapes as geodesics in a volume-preserving diffeomorphism group, showing that minimizing kinetic energy action leads to Euler equations for inviscid potential flow with zero pressure and surface tension. The key result is that any two equal-volume shapes can be approximately connected via an Euler spray of ellipsoidal geodesics, with the infimum action equaling the Wasserstein distance squared—almost never attained except in 1D.

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 be shape densities (characteristic functions). 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.

Motivation & Objective

• To formalize incompressible fluid flows between shapes as critical paths of kinetic energy under volume-preserving constraints.
• To investigate the instability in the action minimization problem linked to microdroplet formation during shape transport.
• To establish a connection between incompressible Euler sprays and weak solutions of the compressible pressureless Euler system.
• To show that Wasserstein geodesics between compactly supported densities arise as limits of incompressible Euler sprays.
• To demonstrate that each Wasserstein geodesic is the unique minimizer of a relaxed least-action principle in a two-fluid mixture model with vacuum.

Proposed method

• Formulates the problem in an Eulerian framework using characteristic functions to represent shape densities.
• Derives formal geodesic equations as the Euler equations for incompressible, inviscid potential flow with zero pressure and surface tension.
• Analyzes the action functional as kinetic energy integrated over transport paths constrained to shape densities.
• Constructs Euler sprays as countable superpositions of ellipsoidal geodesics to approximate shape transitions.
• Applies relaxed least-action principles in a two-fluid mixture theory to characterize Wasserstein geodesics as minimizers.
• Uses weak limit arguments to connect incompressible Euler sprays to solutions of the compressible pressureless Euler system.

Experimental results

Research questions

• RQ1 What are the geodesic equations governing incompressible fluid flows that transport one shape into another while preserving volume?
• RQ2 How does microdroplet formation destabilize the action minimization problem for shape transport?
• RQ3 What is the relationship between Euler sprays and the Wasserstein distance in shape space?
• RQ4 Can Wasserstein geodesics between compactly supported densities be realized as limits of incompressible Euler flows?
• RQ5 Is there a variational principle that uniquely characterizes Wasserstein geodesics in terms of a relaxed action functional?

Key findings

• The infimum of the action for transporting two equal-volume shapes is equal to the square of the Wasserstein distance, and this value is almost never attained except in one dimension.
• Any two shapes of equal volume can be approximately connected by an Euler spray, a countable superposition of ellipsoidal geodesics, demonstrating the existence of approximate minimizers.
• Each Wasserstein geodesic between bounded, compactly supported densities arises as a weak limit of incompressible Euler sprays.
• The resulting weak limit is a solution to the compressible pressureless Euler system, linking incompressible and compressible fluid dynamics via the action principle.
• Wasserstein geodesics are the unique minimizers of a relaxed least-action principle in a two-fluid mixture model involving incompressible fluid and vacuum.
• The formal geodesic equations correspond to the Euler equations for incompressible, inviscid potential flow with zero pressure and surface tension on the free boundary.