Dynamical Systems Methods for Fluid Mechanics and Hamiltonian Mechanics

研究人员研究偏微分方程和其他无限维动力系统的行为，并采用和发展起源于有限维动力学研究的思想，以对这类系统的解的行为作出定性和定量的预测。他主要关注物理应用程序中出现的问题。他研究的问题包括流体系统的亚稳态行为，即在相对较短的时间尺度上，流体系统中出现像漩涡这样的长寿命结构，然后决定流体随后很长时间的演化。他还研究了涡旋等局部化结构对旋转的分层流体层(如大气)行为的影响。微分方程产生于各种不同的物理环境中，其特点是虽然方程本身是众所周知的，但它们太复杂了，除非在非常特殊或物理上不现实的情况下才能显式求解。然而，应用程序至少需要对其解决方案的行为有一个定性的理解；本研究项目旨在发展这样的理解。该项目将研究生和博士后研究员的培训融入到这项研究的各个方面。在研究人员研究的具体系统中，包括：(I)使用正规形和超矫顽力理论的几乎无粘流体和弱阻尼哈密顿系统，如Fermi-Pasta-Ulam系统；(Ii)不变流形及其对以下行为的影响：(A)不稳定分层的旋转流体系统，(B)可压缩的Navier-Stokes方程和(C)作为理解动力学系统连续统近似的手段，以及(Iii)流体力学和其他无限维系统中的小约数。将动力系统思想应用于无界域上的偏微分方程组的一个特点是连续谱和离散谱的相互作用。关于分层流体和动力学系统的近似定理的计划工作进一步发展了不变流形理论来处理这些问题。涡片方程中小因子问题的研究将类KAM方法推广到一类新的无限维方程组。对弱粘性流体和弱阻尼谐振子系统的研究，既加深了对这些系统中中间时间尺度起源的内在理解，又阐明了这两个明显不同的物理系统之间的数学关系。研究生和博士后研究员被包括在该项目的工作中。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.