Analysis of Partial Differential Equations Using Dynamical Systems Techniques

使用动力系统技术分析偏微分方程

• 批准号: 1600061
• 负责人: Margaret Beck
• 金额: $ 0.9万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600061&HistoricalAwards=false
• 关键词: Analysis; Partial; Differential; Equations; Using

This award provides support for participants, especially graduate students, junior researchers, women, and mathematicians from groups under-represented in the sciences, to attend the conference "Analysis of Partial Differential Equations using Dynamical Systems Techniques" hosted by Boston University during June 1-3, 2016. Partial differential equations (PDE) arise as mathematical descriptions of many natural and societal phenomena, for example, processes that change in both space and time, such as fluid mechanics, particle interaction, or financial markets. On the other hand, dynamical systems theory originated in classical Newtonian mechanics and developed into a universal tool for studying time evolution of a very broad range of phenomena. Techniques from the theory of dynamical systems have been used to analyze certain classes of PDE. The purpose of this conference is to bring together researchers who have pioneered the use of such techniques in the context of PDE, in order to share ideas, promote collaboration, and advance our understanding of these PDE models and how dynamical systems techniques can be used to analyze them. In addition, the conference will further our ability to predict dynamics of the physical phenomena that these PDEs describe. Many of the conference participants will be junior researchers, such as PhD students, who will benefit greatly from interacting with more senior researchers and also from presenting their own work. In the last several decades techniques from the theory of finite-dimensional dynamical systems have proven to be extremely useful for analyzing certain classes of PDE that can be viewed as infinite-dimensional dynamical systems. These PDE are viewed as evolution equations whose phase space is an appropriate infinite-dimensional Banach or Hilbert space. Methods such as semigroup theory, invariant manifolds, stability analysis, and KAM theory have all been utilized to great effect for a variety of these PDE, including the Navier-Stokes equation, Korteweg-deVries equation, Boussinesq equation, nonlinear Schrödinger equation, and the Fermi-Pasta-Ulam model. The conference will focus on two main types of PDE where this dynamical systems approach has been particularly successful: Hamiltonian systems and models from fluid dynamics. This focus on the two classes of PDE mentioned above will allow for a more complete coverage of recent related developments. Broader impacts will result from the participation of early-career researchers and those from underrepresented groups, who will be able to network and interact with senior participants and to learn about emerging directions in their field. Furthermore, both Hamiltonian systems and fluids models arise from practical applications, and thus the advancement in our understanding of the behavior of solutions to those models will in turn forward our ability to predict the types of behaviors observed in the physical systems themselves. More information can be found at the conference website: http://math.bu.edu/people/mabeck/APDE-DS.html

该奖项为参与者，特别是研究生，初级研究人员，妇女和数学家的支持提供了支持，这些小组的科学中代表性不足，他们参加了“使用动态系统技术对部分微分方程进行分析”，由波士顿大学在2016年6月1-3日期间举办的动态系统技术”。时间，例如流体力学，粒子互动或金融市场。另一方面，动态系统理论起源于古典牛顿力学，并发展成为一种通用工具，用于研究各种现象的时间演变。动态系统理论的技术已用于分析某些类别的PDE类别。这次会议的目的是将在PDE背景下使用此类技术的使用率汇集在一起​​，以分享想法，促进协作并促进我们对这些PDE模型的理解以及如何使用动态系统技术来分析它们。此外，会议将进一步预测这些PDE所描述的物理现象动态的能力。许多会议参与者将是初级研究人员，例如博士生，他们将与更多的高级研究人员互动以及介绍自己的工作中受益匪浅。在过去的几十年中，有限维动力学系统理论的技术已被证明对分析某些类别的PDE非常有用，而PDE可以将其视为无限二维动力学系统。这些PDE被视为相位空间是适当的无限二维Banach或Hilbert空间的进化方程。诸如Semigroup理论，不变流形，稳定性分析和KAM理论之类的方法都被用于各种PDE的效果，包括Navier-Stokes方程，Korteweg-Devries方程，Boussinesq方程，非线性Schrödinger方程以及Fermi-Pasta-Pasta-Pasta-pasta-ulam模型。会议将重点介绍这种动态系统方法特别成功的两种主要类型：Hamiltonian系统和流体动力学模型。该关注上面提到的两类PDE的重点将允许对最近相关的发展进行更完整的报道。早期研究人员和代表性不足的群体的参与将产生更大的影响，他们将能够与高级参与者建立联系和互动，并了解其领域的新兴方向。此外，汉密尔顿系统和流体模型均来自实际应用，因此，我们对解决方案对这些模型的行为的理解的进步反过来，我们将能够预测物理系统本身观察到的行为类型。更多信息可以在会议网站上找到：http：//math.bu.edu/people/mabeck/apde-ds.html