Analysis of Partial Differential Equations Using Dynamical Systems Techniques

Trustees Of Boston University, Boston

Investigators

Abstract

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