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

Gallouët, Thomas; Vialard, François-Xavier (2018), The Camassa-Holm equation as an incompressible Euler equation: a geometric point of view, Journal of Differential Equations, 264, 7, p. 4199-4234. 10.1016/j.jde.2017.12.008

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Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01363647
Date
2018
Journal name
Journal of Differential Equations
Volume
264
Number
7
Publisher
Elsevier
Pages
4199-4234
Publication identifier
10.1016/j.jde.2017.12.008
Metadata
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Author(s)
Gallouët, Thomas
Centre de Mathématiques Laurent Schwartz [CMLS]
Vialard, François-Xavier
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.
Subjects / Keywords
group of diffeomorphisms; Wasserstein-Fisher-Rao; Camassa-Holm; Optimal transport; Hellinger-Kantorovich; EPDiff; polar factorization

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