Geometric hydrodynamics and Hamiltonian structures

几何流体动力学是数学的一个年轻分支，它通过在无限维配置空间上发展微分几何和辛几何来研究流体运动。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各种物理现象，从大气流动和洋流到托卡马克和恒星的磁场。它们有许多相似之处，包括哈密顿结构、对称性和守恒定律、几何公式等。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（例如湍流的出现和奇点形成）做出贡献。它重点关注以下相关项目。 该提案的第一个目标是极大地扩展最近引入的牛顿方程在微分同胚和概率密度的无限维配置空间上的几何框架。它已经涵盖了几个重要的流体动力学起源的偏微分方程，包括各种可压缩流体和薛定谔型方程。此类方程与相应相空间之间的马德隆变换的卡勒性质表明，量子力学和几何流体动力学之间的联系比以前认识到的要紧密得多。我希望对最近发现的流体动力学量子类似物有新的认识。 研究的另一个方向是将阿诺德的测地线方法从李群扩展到李群群。虽然阿诺德的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，但其适用范围仅限于具有对称群的系统。我将通过使用李群群结构将其扩展到更广泛的流体动力学起源系统，从这种哈密顿方法导出涡片的开尔文-亥姆霍兹不稳定性，研究与形状空间度量的关系，以及使用这种技术以获得具有不连续初始数据的欧拉方程的存在性和唯一性结果。 我还计划扩大五角星图高维推广的可积和不可积族的研究。 Boussinesq 型方程和 KdV 层次结构在此类图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众所周知的流体动力学起源的哈密顿偏微分方程，同时保留其主要结构。我计划为这种离散哈密顿图找到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二维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.