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Zbl 1139.53040
Kolev, Boris
Poisson brackets in hydrodynamics. (English)
[J] Discrete Contin. Dyn. Syst. 19, No. 3, 555-574 (2007). ISSN 1078-0947; ISSN 1553-5231/e

This paper discusses various Poisson structures which provide the Hamiltonian formulations of numerous evolution equations in fluid mechanics. The principal observation is that in infinite dimensions the Lie-Poisson brackets are not defined for all smooth functionals. This rises the following questions: whether the chosen bracket is closed for the class of the functionals for which it is defined and whether the fundamental Jacobi identity is satisfied? The presented discussion based on concrete examples and the supplied list of references will be of definite help for the readers who are interested in peculiarities of Lie-Poisson structures in infinite dimensional spaces.
[Ivailo Mladenov (Sofia)]

MSC 2000:: *53D17 Poisson manifolds
37K05 Hamiltonian structures, etc.
37K65 Hamiltonian systems on groups of diffeomorphisms etc.

Keywords: Lie-Poisson structures; Hamiltonian structures; nonlinear equations; rotational fluid flows

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