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.

该项目旨在开发用于研究流体运动的新数学工具。具有足够的数学语言来描述其描述至关重要，对于理解气象学中空气湍流以及液体和等离子体中的大型和小规模结构等现象至关重要。尽管努力了很多努力，但流体动力学的许多方面仍然很少理解，并且只有在使用各种不同的数学工具时，才有可能的突破似乎是可能的。最有希望的方向之一是流体的几何方法。众所周知，这种方法适用于局限于固定域的流体。该项目的目的是将几何语言扩展到更一般的环境，包括波浪形成，洋流，对涡流不稳定性的洞察力以及水下车辆运动的研究。研究者将积极参与研究生参与该项目。现代的几何流体动力学起源于1960年代，当时Arnold证明了理想流体的Euler方程描述了右转量的右数量的测量流量，该集量是在体积量化域上的右旋数量，而流动域则是流动域的差异性差异。事实证明，这种见解对于研究中的哈密顿特性和保存定律是必不可少的，在流体动力学，流体不稳定性，流的拓扑特性以及获得Euler-type方程的独特性结果的强大工具中是必不可少的。此外，Arnold对不可压缩流体的群体理论描述也已被证明适用于许多其他与流体相关的环境中，包括磁水动力学，可压缩的液体，半地球形液体和Kortewegeg-de Vries方程。但是，阿诺德方法的适用范围仅限于对称形成组的系统。同时，流体动力学中存在许多问题，例如自由边界问题，流体结构相互作用，不连续的流体流以及多相和分层流体，其对称性应被视为群体类似体。该项目的目的是在各种流体动力问题的背景下建立无限尺寸的li子类固定范围，并应用此范式来解决一系列对应用程序感兴趣的具体问题。该奖项反映了NSF的法定任务，并通过评估基金会的范围来评估支持者，并通过基金会的范围进行了评估和广泛的影响。