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SUMMARY:On the 2-point problem for the Lagrange-Euler equation - Shnirelma
 n\, A (Concordia University\, Canada)
DTSTART:20120828T103000Z
DTEND:20120828T113000Z
UID:TALK39419@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Consider the motion of ideal incompressible fluid in a bounded
  domain (or on a compact Riemannian manifold). The configuration space of 
 the fluid is the group of volume-preserving diffeomorphisms of the flow do
 main\, and the flows are geodesics on this infinite-dimensional group wher
 e the metric is defined by the kinetic energy. The geodesic equation is th
 e Lagrange-Euler equation.\nThe problem usually studied is the initial-val
 ue problem\, where we look for a geodesic with given initial fluid configu
 ration and initial velocity field. In this talk we consider a different pr
 oblem: find a geodesic connecting two given fluid configurations. The main
  result is the following\nTheorem: Suppose the flow domain is a 2-dimensio
 nal torus. Then for any two fluid configurations there exists a geodesic c
 onnecting them. This means that\, given arbitrary fluid configuration (dif
 feomorphism)\, we can "push"\nthe fluid along some initial velocity field\
 , so that by time one the fluid\, moving according to the Lagrange- Euler 
 equation\, assumes the given configuration. This theorem looks superficial
 ly like the Hopf-Rinow theorem for finite-dimensional Riemannian manifolds
 . In fact\, these two theorems have almost nothing in common.\nIn our case
 \, unlike the Hopf-Rinow theorem\, the geodesic is not\, in general case\,
  the shortest curve connecting the endpoints (fluid configurations).\nMore
 over\, the length minimizing curve can not exist at all\, while the geodes
 ic always exists. The proof is based on some ideas of global analysis (Fre
 dholm quasilinear maps) and microlocal analysis of the Lagrange-Euler equa
 tion (which may be called a ?microglobal analysis?).\n\n
LOCATION:Seminar Room 1\, Newton Institute
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