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dc.contributor.authorGallouët, Thomas
dc.contributor.authorVialard, François-Xavier
dc.date.accessioned2018-09-04T13:36:20Z
dc.date.available2018-09-04T13:36:20Z
dc.date.issued2018
dc.identifier.issn0022-0396
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/17944
dc.language.isoenen
dc.subjectgroup of diffeomorphismsen
dc.subjectWasserstein-Fisher-Raoen
dc.subjectCamassa-Holmen
dc.subjectOptimal transporten
dc.subjectHellinger-Kantorovichen
dc.subjectEPDiffen
dc.subjectpolar factorizationen
dc.subject.ddc511en
dc.titleThe Camassa-Holm equation as an incompressible Euler equation: a geometric point of viewen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThe 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.en
dc.relation.isversionofjnlnameJournal of Differential Equations
dc.relation.isversionofjnlvol264en
dc.relation.isversionofjnlissue7en
dc.relation.isversionofjnldate2018
dc.relation.isversionofjnlpages4199-4234en
dc.relation.isversionofdoi10.1016/j.jde.2017.12.008en
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-01363647en
dc.relation.isversionofjnlpublisherElsevieren
dc.subject.ddclabelPrincipes généraux des mathématiquesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2018-09-04T13:34:07Z
hal.person.labIds18
hal.person.labIds60


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