The Lie-Poisson structure of the Euler equations of an ideal fluid

Vasylkevych, S. and J. E. Marsden

Dynamics of PDE 2, 281-300

Abstract:

This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in $\mathbb {R}$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 C¹ 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.