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Lie Groupoids and Infinite-Dimensional Dynamical Systems

李群群和无限维动力系统

基本信息

批准号：
2008021
负责人：
Anton Izosimov
金额：
$ 21.7万
依托单位：
University of Arizona
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2008021&HistoricalAwards=false
关键词：
Lie Groupoids Infinite Dimensional Dynamical

项目摘要

This project aims at developing new mathematical tools for studying the motion of fluids. Having an adequate mathematical language for their description is crucial for understanding such phenomena as formation of air turbulence in meteorology, as well as large- and small-scale structures in liquids and plasmas. Despite much effort, many aspects of fluid dynamics are still poorly understood and a breakthrough in this field only seems possible if a variety of different mathematical tools is used. One of the most promising directions is a geometric approach to fluids. This approach is known to work well for fluids confined to a fixed domain. The goal of the project is to extend the geometric language to more general settings, with applications including formation of waves, ocean currents, insight into vortex instabilities, and the study of motion of underwater vehicles. The investigator will actively involve graduate students in this project.Modern geometric fluid dynamics originated in the 1960s when V. Arnold proved that the Euler equation for an ideal fluid describes the geodesic flow of a right-invariant metric on the group of volume-preserving diffeomorphisms of the flow domain. This insight turned out to be indispensable for the study of Hamiltonian properties and conservation laws in hydrodynamics, fluid instabilities, topological properties of flows, as well as a powerful tool for obtaining sharper existence and uniqueness results for Euler-type equations. Furthermore, Arnold's group-theoretic description of incompressible fluids has also been shown to be applicable in many other fluid-related settings, including magnetohydrodynamics, compressible fluids, semi-geostrophic and the Korteweg-de Vries equations. However, the scope of applicability of Arnold's approach is limited to systems whose symmetries form a group. At the same time, there are many problems in fluid dynamics, such as free boundary problems, fluid-structure interactions, discontinuous fluid flows, as well as multiphase and stratified fluids, whose symmetries should instead be regarded as a groupoid. The aim of the project is to develop a paradigm of infinite-dimensional Lie groupoids in the context of various fluid-dynamical problems, as well as to apply this paradigm to approach a range of concrete questions that are of interest for applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目旨在开发新的数学工具来研究流体的运动。有一个适当的数学语言来描述它们，对于理解气象中空气湍流的形成以及液体和等离子体中的大小尺度结构等现象至关重要。尽管做了很多努力，流体动力学的许多方面仍然知之甚少，只有使用各种不同的数学工具，才有可能在这一领域取得突破。最有希望的方向之一是对流体进行几何处理。众所周知，这种方法对限制在固定区域内的流体很有效。该项目的目标是将几何语言扩展到更一般的环境，应用包括波浪形成、洋流、对涡旋不稳定性的洞察以及对水下航行器运动的研究。现代几何流体力学起源于20世纪60年代，当时V.Arnold证明了理想流体的欧拉方程描述了流域的保体积微分同胚群上右不变度规的测地线流动。这对于研究流体动力学中的哈密顿性质和守恒律、流体不稳定性和流动的拓扑性质是必不可少的，也是获得欧拉型方程更精确的存在唯一性结果的有力工具。此外，Arnold关于不可压缩流体的群论描述也被证明适用于许多其他与流体有关的环境，包括磁流体动力学、可压缩流体、半地转流体和Korteweg-de Vries方程。然而，Arnold方法的适用范围仅限于对称性为一组的系统。同时，流体动力学中存在许多问题，如自由边界问题、流固耦合问题、不连续流体流动问题、多相流体和分层流体问题等，这些问题的对称性应该被看作是群体。该项目的目的是在各种流体动力学问题的背景下发展无限维李群胚的范例，并应用该范例来处理一系列感兴趣的具体应用问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。