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# Dynamics of Partial Differential Equations

## Volume 2 (2005)

### Number 4

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

Pages: 281 – 300

DOI: http://dx.doi.org/10.4310/DPDE.2005.v2.n4.a1

#### Authors

#### Abstract

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 H^{s} 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.

#### Keywords

Euler equations, Poisson map, Lie-Poisson bracket, Lagrangian representation, Lie-Poisson reduction procedure

#### 2010 Mathematics Subject Classification

Primary 35-xx. Secondary 76-xx.