Analysis of Partial Differential Equations Using Dynamical Systems Techniques

该奖项为参加波士顿大学于2016年6月1日至3日举办的“使用动力系统技术分析偏微分方程”会议的参与者，特别是研究生、初级研究人员、女性和数学家提供支持。偏微分方程（PDE）作为对许多自然和社会现象的数学描述而出现，例如，在空间和时间上变化的过程，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。来自动力系统理论的技术已被用于分析某些类型的偏微分方程。本次会议的目的是将在PDE背景下率先使用这些技术的研究人员聚集在一起，以便分享想法，促进合作，并推进我们对这些PDE模型的理解以及如何使用动力系统技术来分析它们。此外，会议将进一步提高我们预测这些偏微分方程所描述的物理现象动力学的能力。许多会议参与者将是初级研究人员，例如博士生，他们将从与更多高级研究人员的互动以及展示自己的工作中受益匪浅。在过去的几十年里，有限维动力系统理论的技术已经被证明对分析某些可以被视为无限维动力系统的偏微分方程非常有用。这些偏微分方程被视为演化方程，其相空间是适当的无限维巴拿赫或希尔伯特空间。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PDE，包括Navier-Stokes方程、Korteweg-deVries方程、Boussinesq方程、非线性Schrödinger方程和Fermi-Pasta-Ulam模型。会议将集中讨论两种主要的PDE类型，其中这种动力系统方法特别成功：哈密顿系统和流体动力学模型。把重点放在上面提到的两类PDE上，可以更全面地介绍最近的相关发展。更广泛的影响将来自早期职业研究人员和那些来自代表性不足群体的研究人员的参与，他们将能够与资深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于实际应用，因此，我们对这些模型解的行为的理解的进步将反过来促进我们预测物理系统本身所观察到的行为类型的能力。更多信息请访问会议网站：http://math.bu.edu/people/mabeck/APDE-DS.html

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