• français
    • English
  • français 
    • français
    • English
  • Connexion
JavaScript is disabled for your browser. Some features of this site may not work without it.
Accueil

Afficher

Cette collectionPar Date de CréationAuteursTitresSujetsNoms de revueToute la baseCentres de recherche & CollectionsPar Date de CréationAuteursTitresSujetsNoms de revue

Mon compte

Connexion

Statistiques

Afficher les statistiques d'usage

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

Thumbnail
Ouvrir
Final.pdf (454.3Kb)
Date
2018
Lien vers un document non conservé dans cette base
https://hal.archives-ouvertes.fr/hal-01363647
Indexation documentaire
Principes généraux des mathématiques
Subject
group of diffeomorphisms; Wasserstein-Fisher-Rao; Camassa-Holm; Optimal transport; Hellinger-Kantorovich; EPDiff; polar factorization
Nom de la revue
Journal of Differential Equations
Volume
264
Numéro
7
Date de publication
2018
Pages article
4199-4234
Nom de l'éditeur
Elsevier
DOI
http://dx.doi.org/10.1016/j.jde.2017.12.008
URI
https://basepub.dauphine.fr/handle/123456789/17944
Collections
  • CEREMADE : Publications
Métadonnées
Afficher la notice complète
Auteur
Gallouët, Thomas
18 Centre de Mathématiques Laurent Schwartz [CMLS]
Vialard, François-Xavier
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Résumé en anglais
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.

  • Accueil Bibliothèque
  • Site de l'Université Paris-Dauphine
  • Contact
SCD Paris Dauphine - Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16

 Cette création est mise à disposition sous un contrat Creative Commons.