Geometric hydrodynamics and Hamiltonian structures

几何流体力学是一门年轻的数学分支，它是在无穷维位形空间上发展微分几何和辛几何来研究流体运动的。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中的磁场。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几何和群论方法来解决流体动力学的关键问题，如湍流的出现和奇点的形成。它侧重于以下相关项目。 该提案的第一个目标是极大地扩展最近引入的几何框架的牛顿方程的无限维配置空间的同构和概率密度。它已经包含了几个重要的偏微分方程的流体动力学起源，包括各种可压缩流体和薛定谔型方程。在这些方程和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认识到的要紧密得多。我希望对最近发现的流体力学量子类似物有新的了解。 另一个研究方向是将Arnold的测地线方法从李群推广到李群胚。虽然阿诺德的方法是必不可少的研究守恒定律和稳定性的流体动力学，其适用范围仅限于系统的对称群。我将扩展到一个更广泛的一类系统的流体动力学起源使用李群胚结构，推导出开尔文-亥姆霍兹不稳定性的涡面从这个哈密顿方法，研究的关系度量的形状空间，以及使用这种技术，以获得存在性和唯一性结果的欧拉方程的不连续的初始数据。 我还计划扩大研究的可积和不可积家庭的高维推广的五角星形地图。Boussinesq型方程和KdV层次结构在连续的限制，这样的地图的外观表示一个几何自然的方式来离散那些著名的哈密顿偏微分方程的流体动力学起源，同时保持其主要结构。我计划找到一个精确的边界的可积性和不可积的离散哈密顿映射。一个相关的更基本的目标是一个严格的证明不可积的二维理想流体力学。

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.