The Camassa-Holm equation as an incompressible Euler equation: a geometric point of view

TL;DR

This paper explores the geometric relationship between the Camassa-Holm equation and incompressible Euler equations through diffeomorphism groups, optimal transport, and fiber bundle automorphisms, revealing new formulations and properties.

Contribution

It establishes a geometric framework linking the Camassa-Holm equation to incompressible Euler equations via diffeomorphism groups and optimal transport theory, including a polar factorization theorem.

Findings

Provides an isometric embedding of diffeomorphism groups into automorphisms of half-density bundles.

Reformulates the Camassa-Holm equation as a geodesic on an automorphism subgroup.

Shows smooth solutions are length minimizing geodesics for short times.

Abstract

The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein-Fisher-Rao distance is a natural extension of the classical L^2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the Hdiv right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa-Holm (CH) equation in one dimension. On the optimal transport side, we prove a polar factorization theorem on the automorphism group of half-densities.Geometrically, our point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L^2 type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On S1, solutions to the standard CH thus give particular solutions of the incompressible Euler equation on a group of homeomorphisms of R^2 which preserve a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler-Arnold equation for the Hdiv right-invariant metric are length minimizing geodesics for sufficiently short times.