The Camassa–Holm equation as an incompressible Euler equation: A geometric point of view.

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 H div right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa–Holm (CH) equation in one dimension. Geometrically, we present 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 S 1 , solutions to the standard CH thus give radially 1-homogeneous solutions of the incompressible Euler equation on R 2 which preserves 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 H div right-invariant metric are length minimizing geodesics for sufficiently short times. [ABSTRACT FROM AUTHOR]