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

Jerrold E. Marsden (Control and Dynamical Systems Department, California Institute of Technology, Pasadena, Calif.)

Sergiy Vasylkevych (Départment de Mathématiques, École Polythechnique Fédérale de Lausanne, Switzerland)

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 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.

Keywords

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

2010 Mathematics Subject Classification

Primary 35-xx. Secondary 76-xx.

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