TL;DR
This paper reviews various Poisson brackets used in fluid mechanics to formulate Hamiltonian structures, focusing on their mathematical rigor and applicability to rotational flows with free boundaries, highlighting unresolved challenges.
Contribution
It analyzes existing Poisson structures in hydrodynamics, discusses their mathematical difficulties, and emphasizes open problems in defining valid brackets for complex fluid flows.
Findings
Identifies key Poisson brackets used in fluid mechanics.
Highlights mathematical difficulties in defining rigorous brackets.
Points out unresolved issues in Hamiltonian formulations for free boundary flows.
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.
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