Dynamics of Partial Differential Equations
Volume 2 (2005)
The Lie-Poisson structure of the Euler equations of an ideal fluid
Pages: 281 – 300
This paper provides a precise sense in which the time t map for the Eulerequations of an ideal fluid in a region in Rⁿ (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 Hs to itself (where s > (n ∕ 2) + 1). The idea of how this diffculty 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.
Euler equations, Poisson map, Lie-Poisson bracket, Lagrangian representation, Lie-Poisson reduction procedure
2010 Mathematics Subject Classification
Primary 35-xx. Secondary 76-xx.