The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid

Details The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid

2007-11-30

arxiv.org/abs/0711.4875v1

Dynamics of PDE, Vol.2, No.4, 281-300, 2005

Mathematics

Analysis of PDEs

24 pages

Scientific paper

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.

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