Dynamics of Partial Differential Equations

Volume 2 (2005)

Number 1

The Lie-Poisson structure of the LAE-α equation

Pages: 25 – 57

DOI: http://dx.doi.org/10.4310/DPDE.2005.v2.n1.a2

Authors

François Gay-Balmaz (Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland)

Tudor S. Ratiu (Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland)

Abstract

This paper shows that the time t map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed boundary conditions is smooth, a result already known for Dirichlet boundary conditions.

Keywords

averaged Euler equation, Dirichlet, Neumann, and mixed boundary condition, Sobolev diffeomorphsim group, Lie-Poisson reduction, geodesic spray

2010 Mathematics Subject Classification

Primary 35Q35, 35Q53, 53D17, 53D22, 53D25. Secondary 58B20, 58B25, 58D05.

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