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

TL;DR

This paper demonstrates that the time evolution maps of the Euler equations for an ideal fluid can be viewed as Poisson maps within the Lie-Poisson framework, despite regularity challenges in the Eulerian description, by leveraging Lagrangian smoothness and reduction techniques.

Contribution

It establishes the Poisson nature of the Euler equations' flow maps using Lie-Poisson reduction, overcoming regularity issues in the Eulerian formulation.

Findings

Time t maps are Poisson maps relative to the Lie-Poisson bracket.

Regularity issues in Eulerian representation are addressed via Lagrangian smoothness.

The approach clarifies the geometric structure underlying ideal fluid dynamics.

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 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.