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SUMMARY:From unbalanced optimal transport to the Camassa-Holm equation
DTSTART:20161207T083000Z
DTEND:20161207T092000Z
DTSTAMP:20241015T003500Z
UID:indico-event-1745@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: François-Xavier VIALARD (Université Paris-Dauphine
)\n\nWe present an extension of the Wasserstein L2 distance to the space o
f positive Radon measures as an infimal convolution between the Wasserstei
n L2 metric and the Fisher-Rao metric. In the work of Brenier\, optimal tr
ansport has been developed in its study of the incompressible Euler equati
on. For the Wasserstein-Fisher-Rao metric\, the corresponding fluid dynami
c equation is known as the Camassa-Holm equation (at least in dimension 1)
\, originally introduced as a geodesic flow on the group of diffeomorphism
s. This point of view provides an isometric embedding of the group of diff
eomorphisms endowed with this right-invariant metric in the automorphisms
group of the fiber bundle of half densities endowed with an L 2 type of co
ne metric. As a direct consequence\, we describe a new polar factorization
on the automorphism group of half-densities which can be seen as a constr
ained version of Brenier’s theorem. The main application consists in wri
ting the Camassa-Holm equation on S^1 as a particular case of the incompre
ssible Euler equation on a group of homeomorphisms of R^2 that preserve a
radial density which has a singularity at 0\, the cone point.\n\nhttps://i
ndico.math.cnrs.fr/event/1745/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/1745/
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