%0 Journal Article
%T The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid
%A Sergiy Vasylkevych
%A Jerrold E. Marsden
%J Mathematics
%D 2007
%I arXiv
%X This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C^1 from the Sobolev class H^s to itself (where s > (n/2) + 1). The idea of how this difficulty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure.
%U http://arxiv.org/abs/0711.4875v1