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SUMMARY:From unbalanced optimal transport to the Camassa-Holm equation
DTSTART;VALUE=DATE-TIME:20161207T083000Z
DTEND;VALUE=DATE-TIME:20161207T092000Z
DTSTAMP;VALUE=DATE-TIME:20210412T041200Z
UID:indico-event-1745@indico.math.cnrs.fr
DESCRIPTION:We present an extension of the Wasserstein L2 distance to the
space of positive Radon measures as an infimal convolution between the Was
serstein L2 metric and the Fisher-Rao metric. In the work of Brenier\, opt
imal transport has been developed in its study of the incompressible Euler
equation. For the Wasserstein-Fisher-Rao metric\, the corresponding fluid
dynamic equation is known as the Camassa-Holm equation (at least in dimen
sion 1)\, originally introduced as a geodesic flow on the group of diffeom
orphisms. This point of view provides an isometric embedding of the group
of diffeomorphisms endowed with this right-invariant metric in the automor
phisms group of the fiber bundle of half densities endowed with an L 2 typ
e of cone metric. As a direct consequence\, we describe a new polar factor
ization on the automorphism group of half-densities which can be seen as a
constrained version of Brenier’s theorem. The main application consists
in writing the Camassa-Holm equation on S^1 as a particular case of the i
ncompressible Euler equation on a group of homeomorphisms of R^2 that pres
erve a radial density which has a singularity at 0\, the cone point.\n\nht
tps://indico.math.cnrs.fr/event/1745/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/1745/
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