Dynamics of Partial Differential Equations
Volume 2 (2005)
The Lie-Poisson structure of the LAE-α equation
Pages: 25 – 57
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.
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.