Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Poisson brackets in Hydrodynamics

Pages: 555 - 574, Volume 19, Issue 3, November 2007

doi:10.3934/dcds.2007.19.555       Abstract        Full Text (251.4K)       Related Articles

Boris Kolev - LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13, France (email)

Abstract: This paper investigates different Poisson structures that have been proposed to give a Hamiltonian formulation to evolution equations issued from fluid mechanics. Our aim is to explore the main brackets which have been proposed and to discuss the difficulties which arise when one tries to give a rigorous meaning to these brackets. Our main interest is in the definition of a valid and usable bracket to study rotational fluid flows with a free boundary. We discuss some results which have emerged in the literature to solve some of the difficulties that arise. It appears to the author that the main problems are still open.

Keywords:  Hamiltonian structures, Hamiltonian systems on groups of diffeomorphisms.
Mathematics Subject Classification:  Primary: 53D17, 37K05; Secondary: 53D20, 37K65.

Received: June 2007;      Revised: July 2007;      Published: July 2007.