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

该奖项为参与者提供支持，特别是研究生，初级研究人员，女性和来自科学界代表不足的群体的数学家，参加2016年6月1日至3日由波士顿大学主办的“使用动态系统技术分析偏微分方程”会议。偏微分方程（PDE）是许多自然和社会现象的数学描述，例如，在空间和时间上都发生变化的过程，如流体力学，粒子相互作用或金融市场。另一方面，动力系统理论起源于经典牛顿力学，并发展成为研究非常广泛现象的时间演化的通用工具。动力系统理论的技术已经被用来分析某些类的偏微分方程。本次会议的目的是汇集研究人员谁率先在PDE的背景下使用这些技术，以分享思想，促进合作，并推进我们对这些PDE模型的理解，以及如何动态系统技术可以用来分析它们。此外，会议将进一步提高我们预测这些偏微分方程描述的物理现象的动力学的能力。许多与会者将是初级研究人员，如博士生，他们将从与更资深的研究人员的互动中受益匪浅，并展示自己的工作。在过去的几十年中，有限维动力系统理论的技术已经被证明是非常有用的分析某些类的PDE，可以被视为无限维动力系统。这些偏微分方程被视为发展方程，其相空间是一个适当的无限维Banach或Hilbert空间。诸如半群理论、不变流形、稳定性分析和KAM理论等方法都被用于各种偏微分方程，包括Navier-Stokes方程、Korteweg-deVries方程、Boussinesq方程、非线性薛定谔方程和Fermi-Pasta-Ulam模型。会议将重点关注两种主要类型的偏微分方程，这种动力系统方法在其中特别成功：汉密尔顿系统和流体动力学模型。将重点放在上述两类偏微分方程上，可以更全面地涵盖最近的相关发展。更广泛的影响将来自早期职业研究人员和代表性不足群体的参与，他们将能够与高级参与者建立网络和互动，并了解其领域的新兴方向。此外，哈密顿系统和流体模型都来自实际应用，因此我们对这些模型解的行为的理解的进步反过来又会提高我们预测物理系统本身所观察到的行为类型的能力。 更多信息可以在会议网站上找到：http://math.bu.edu/people/mabeck/APDE-DS.html