Generalized compressible flows and solutions of the H(div) geodesic problem

TL;DR

This paper explores the geodesic problem on the diffeomorphism group with H(div) metric, proposing a relaxation approach to analyze solutions, proving their minimality, and illustrating the connection to fluid dynamics through numerical experiments.

Contribution

It introduces a Brenier-type relaxation for the H(div) geodesic problem, establishes existence and uniqueness results, and develops a numerical scheme for generalized solutions.

Findings

Smooth H(div) geodesics are globally length minimizing for short times.

A unique pressure field exists for the relaxed solutions.

Numerical results demonstrate the relation between generalized Camassa-Holm and Euler solutions.

Abstract

We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible multi-dimensional generalizations when d>1. We propose a relaxation {\`a} la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.