Dynamical Systems Methods for Fluid Mechanics and Hamiltonian Mechanics

流体力学和哈密顿力学的动力系统方法

基本信息

批准号：
1813384
负责人：
Clarence Wayne
金额：
$ 25.55万
依托单位：
Trustees of Boston University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-10-01 至 2023-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1813384&HistoricalAwards=false
关键词：
Dynamical Systems Methods Fluid Mechanics

项目摘要

The investigator studies the behavior of partial differential equations and other infinite dimensional dynamical systems, and adapts and develops ideas originating in the study of finite-dimensional dynamics to make qualitative and quantitative predictions about the behavior of solutions of such systems. He focuses primarily on questions arising in physical applications. Among the questions he studies are the metastable behavior of fluid systems, whereby long-lived structures like vortices appear in the flow on a relatively short time scale, and then determine the subsequent evolution of the fluid for very long times. He also studies the influence of localized structures like vortices on the behavior of rotating, stratified fluid layers like the atmosphere. The differential equations arise in a variety of different physical contexts and are characterized by the fact that while the equations themselves are well known, they are too complicated to solve explicitly except in very special or physically unrealistic cases. Nevertheless, applications require at least a qualitative understanding of the behavior of their solutions; this research project aims to develop such an understanding. The project incorporates the training of graduate students and post-doctoral fellows into all facets of this research.Among the specific systems that the investigator studies are: (i) nearly inviscid fluids and weakly damped Hamiltonian systems like the Fermi-Pasta-Ulam system using normal forms and the theory of hypercoercivity; (ii) invariant manifolds and their implications for the behavior of: (a) unstably stratified, rotating fluid systems, (b) the compressible Navier-Stokes equations and (c) as a means of understanding the continuum approximation of kinetic systems, and (iii) small-divisors in fluid mechanics and other infinite dimensional systems. One feature that makes it difficult to apply dynamical systems ideas to partial differential equations on unbounded domains is the interplay of effects due to continuous spectrum with those coming from discrete spectrum. The planned work on stratified fluids and approximation theorems for kinetic systems further develops the theory of invariant manifolds to treat these problems. The study of small divisor problems in the vortex sheet equations extends KAM-like methods to a new class of infinite dimensional systems. The study of weakly viscous fluids and weakly damped oscillator systems both leads to a deeper intrinsic understanding of the origin of intermediate time scales in these systems and illuminates the mathematical relationships between these two apparently quite different physical systems. Graduate students and post-doctoral fellows are included in the work of the project.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.

研究者研究了部分微分方程和其他无限尺寸动力学系统的行为，并适应和发展源于有限维动力学研究的思想，以对这种系统的解决方案进行定性和定量预测。他主要专注于物理应用中引起的问题。他研究的问题包括流体系统的亚稳定性行为，从而以相对较短的时间尺度出现了涡流等长期结构，然后确定很长时间的流体演变。他还研究了诸如涡流之类的局部结构对旋转，分层流体层（如大气）的行为的影响。微分方程出现在各种不同的物理环境中，其特征是，尽管方程本身是众所周知的，但它们太复杂了，无法明确解决，除非在非常特殊或身体上不切实际的情况下。然而，应用至少需要对解决方案行为的定性理解；该研究项目旨在发展这种理解。该项目将研究生和博士后研究员的培训纳入了这项研究的所有方面。研究人员研究的特定系统是：（i）使用正常形式和超高效率的理论，几乎是无粘性的液体和弱化的汉密尔顿系统（例如费米 - 帕斯塔 - 乌拉姆系统）等弱化的汉密尔顿系统；（ii）不变的歧管及其对以下行为的影响：（a）不及时的旋转流体系统，（b）可压缩的navier-stokes方程，以及（c）作为理解动态系统的连续近似值的手段，以及（iii）在流体机械和其他无限的地上系统中的小探测器。一项功能使很难将动态系统思想应用于无界域的部分微分方程是由于连续频谱与来自离散频谱的频谱的相互作用。动力学系统的分层流体和近似定理的计划工作进一步发展了不变的歧管理论来治疗这些问题。对涡旋方程中小型除数问题的研究将类似KAM的方法扩展到新的无限尺寸系统。对弱粘性液体和弱阻尼振荡器系统的研究都使人们对这些系统中中间时间尺度的起源有了更深入的内在理解，并阐明了这两个显然完全不同的物理系统之间的数学关系。该项目的工作中包括研究生和博士后研究员。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被视为值得通过评估来获得支持。