Geometric hydrodynamics and Hamiltonian structures

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Geometric hydrodynamics is a young branch of mathematics which studies fluid motion by developing the differential and symplectic geometry on infinite-dimensional configuration spaces. The hydrodynamical Euler equations for compressible and incompressible fluids, as well as their relatives, describe a variety of physical phenomena, from atmospheric flows and ocean currents to magnetic fields in tokamaks and stars. They share many similarities including Hamiltonian structures, symmetries and conservation laws, geometric formulation, etc. The proposed research program is aimed to contribute toward solution of key problems of fluid dynamics, such as the emergence of turbulence and singularity formation, by exploring the geometric and group-theoretical approaches to fluids. It focuses on the following related projects. ******The proposal's first goal is to vastly expand the recently introduced geometric framework of Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and probability densities. It already encompasses several important PDEs of hydrodynamical origin, including various compressible fluids and the Schroedinger-type equations. The discovered Kahler property of the Madelung transform between such equations and corresponding phase spaces indicated a much closer connection of quantum mechanics and geometric hydrodynamics than was previously recognized. I hope to shed new light on recently discovered hydrodynamical quantum analogues.******Another direction of research is to extend Arnold's geodesic approach from Lie groups to Lie groupoids. While Arnold's approach is indispensable for the study of conservation laws and stability in hydrodynamics, its scope of applicability was limited to systems with symmetry groups. I am going to extend this to a much broader class of systems of hydrodynamical origin by using Lie groupoid structures, to derive the Kelvin-Helmholtz instabilities of vortex sheets from this Hamiltonian approach, to study a relation to metrics on shape spaces, as well as to use this technique in order to obtain existence and uniqueness results for the Euler equation with discontinuous initial data.******I also plan to broaden the study of integrable and non-integrable families of higher-dimensional generalizations of pentagram maps. The appearance of Boussinesq-type equations and the KdV hierarchy in continuous limits of such maps indicates a geometrically natural way to discretize those well-known Hamiltonian PDEs of hydrodynamical origin, while preserving their main structures. I plan to find a precise border of integrability and non-integrability for such discrete Hamiltonian maps. A related more fundamental goal is a rigorous proof of non-integrability of 2D ideal hydrodynamics.