Geometric hydrodynamics and Hamiltonian structures

几何流体力学是通过发展无穷维位形空间上的微分几何和辛几何来研究流体运动的一门年轻的数学分支。可压缩和不可压缩流体的流体动力学欧拉方程及其相关方程描述了从大气流动和洋流到托卡马克和恒星中的磁场的各种物理现象。它们有许多相似之处，包括哈密顿结构、对称性和守恒定律、几何表述等。拟议的研究计划旨在通过探索流体的几何和群论方法，为解决流体动力学的关键问题，如湍流的出现和奇点的形成做出贡献。它侧重于以下相关项目。*该提议的第一个目标是极大地扩展最近引入的牛顿方程在具有微分同胚和概率密度的无限维位形空间上的几何框架。它已经包含了几个重要的流体力学起源的偏微分方程组，包括各种可压缩流体和薛定谔类型的方程。在这些方程和相应的相空间之间发现的马德隆变换的卡勒性质表明，量子力学和几何流体力学之间的联系比以前所认识的要紧密得多。我希望对最近发现的流体动力学量子类似物有新的了解。*另一个研究方向是将Arnold的测地线方法从Lie群扩展到Lie群组。虽然Arnold方法对于研究流体力学中的守恒定律和稳定性是不可或缺的，但它的适用范围仅限于具有对称群的系统。我将利用Lie群结构将其推广到更广泛的流体力学起源系统，从这种哈密顿方法推导涡旋片的Kelvin-Helmholtz不稳定性，研究与形状空间上的度规的关系，以及使用这种方法来获得具有不连续初值的欧拉方程的存在唯一性结果。*我还计划扩大对五星映射的高维推广的可积族和不可积族的研究。Boussinesq型方程和KdV方程在这类映射的连续极限中的出现表明了一种几何上自然的方法来离散那些著名的流体力学起源的哈密顿偏微分方程组，同时保持它们的主要结构。我计划找出这种离散哈密顿映射的可积性和不可积性的精确边界。一个相关的更基本的目标是严格证明2D理想流体力学的不可积性。

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.