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Type of Document	Dissertation
Author	Vasylkevych, Sergiy
Author's Email Address	sergiy_vas AT fastmail.fm
URN	etd-05212004-052003
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-05212004-052003
Title	Poisson structures for PDEs associated with diffeomorphism groups
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title J. E. Marsden Committee Chair A. Borodin Committee Member S. Denissov Committee Member V. Kaloshin Committee Member
Keywords	infinitedimensional manifold Hamiltonian dynamics geometric PDE
Date of Defense	2004-04-09
Availability	unrestricted
Abstract We study Poisson and Lie-Poisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear PDEs. In particular, we provide a precise analytic sense in which the time t map for the Euler equations of an ideal fluid in a region of R^n (or on a smooth compact n-manifold with a boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphisms. The key difficulty in finding a suitable context for that arises from the fact that the integral curves of Euler equations are not differentiable on the Lie algebra of divergence free vector fields of Sobolev class Hs. We overcome this obstacle by utilizing the smoothness that one has in Lagrangian representation and carefully performing a non-smooth Lie-Poisson reduction procedure on the appropriate functional classes. This technique is generalized to an arbitrary diffeomorphism group possessing a smooth spray. The applications include the Camassa-Holm equation on S^1, the averaged Euler and EPDiff equations on the n-manifold with a boundary. In all cases we prove that time t map is Poisson on the appropriate Lie algebra of Hs vector fields, where s > n/2 + 1 for the Euler equation and s > n/2 + 2 otherwise.
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