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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:On the 2-point problem for the Lagrange-Euler equa
tion - Shnirelman\, A (Concordia University\, Cana
da)
DTSTART;TZID=Europe/London:20120828T113000
DTEND;TZID=Europe/London:20120828T123000
UID:TALK39419AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/39419
DESCRIPTION:Consider the motion of ideal incompressible fluid
in a bounded domain (or on a compact Riemannian ma
nifold). The configuration space of the fluid is t
he group of volume-preserving diffeomorphisms of t
he flow domain\, and the flows are geodesics on th
is infinite-dimensional group where the metric is
defined by the kinetic energy. The geodesic equati
on is the Lagrange-Euler equation.\nThe problem us
ually studied is the initial-value problem\, where
we look for a geodesic with given initial fluid c
onfiguration and initial velocity field. In this t
alk we consider a different problem: find a geodes
ic connecting two given fluid configurations. The
main result is the following\nTheorem: Suppose the
flow domain is a 2-dimensional torus. Then for an
y two fluid configurations there exists a geodesic
connecting them. This means that\, given arbitrar
y fluid configuration (diffeomorphism)\, we can "p
ush"\nthe fluid along some initial velocity field\
, so that by time one the fluid\, moving according
to the Lagrange- Euler equation\, assumes the giv
en configuration. This theorem looks superficially
like the Hopf-Rinow theorem for finite-dimensiona
l Riemannian manifolds. In fact\, these two theore
ms have almost nothing in common.\nIn our case\, u
nlike the Hopf-Rinow theorem\, the geodesic is not
\, in general case\, the shortest curve connecting
the endpoints (fluid configurations).\nMoreover\,
the length minimizing curve can not exist at all\
, while the geodesic always exists. The proof is b
ased on some ideas of global analysis (Fredholm qu
asilinear maps) and microlocal analysis of the Lag
range-Euler equation (which may be called a ?micro
global analysis?).\n\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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