Hamiltonian partial differential equations and dynamical systems, and their applications

哈密​​顿偏微分方程和动力系统及其应用

• 批准号: RGPIN-2016-05473
• 负责人: Craig, Walter
• 金额: $ 1.25万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675862
• 关键词: Hamiltonian; partial; differential; equations; dynamical

My research program involves the analysis of solutions of partial differential equations (PDEs) that describe wave evolution in a broad spectrum of physical phenomena. I am working on problems of ocean wave dynamics, techniques of Hamiltonian mechanics and their applications to vortex dynamics, nonlinear Schrodinger equations including ones relevant to photonics, and many other problems in nonlinear waves. My research mostly involves theoretical analysis,***contributing to the understanding of PDEs. However in the past it has had an impact on other disciplines, where advances lend themselves to more efficient numerical simulations or to a deeper conceptual understanding of experiments. And I often collaborate with physical scientists on projects in areas having to do with wave phenomena. *** *This proposal is for research on the properties of solutions of specific PDEs that arise in the physical sciences, which describe wave phenomena such as free surface water waves, nonlinear optics, and continuum mechanics. This includes dispersive evolution equations such as the nonlinear Schrodinger and Klein - Gordon equations, vortex filament evolution for the Euler equations, and problems of water waves. I also plan to continue my prior analysis of the incompressible Navier - Stokes equations.*** ***This program has three principal objectives: ***(1) To address Hamiltonian PDEs from the point of view of a phase space analysis, extending the refined analytic techniques of Hamiltonian systems, such as KAM theory, normal forms, and Nekhoroshev stability, to the dynamics of PDEs in the appropriate infinite dimensional setting in which they are naturally posed.***(2) To study free surface water waves in several settings, and to use their property of a Hamiltonian PDE to describe nonlinear interactions, solitary wave collisions, and asymptotic scaling regimes such as using homogenization techniques to describe waves over a variable bathymetry.***(3) To consider aspects of regularity theory for the Navier - Stokes equations, including the ramifications of upper bounds on spectral behavior, and the recent microlocal lower bounds on the singular set.***I always involve postdoctoral fellows and graduate students in my research projects. In addition, I am intending to address problems that have a potential value to other areas of science, and I plan to continue to seek out interesting collaborations with non-mathematician physical scientists, as I have successfully pursued in the past.**

我的研究项目涉及偏微分方程 (PDE) 解的分析，这些方程描述了广泛的物理现象中的波演化。我正在研究海浪动力学问题、哈密顿力学技术及其在涡动力学中的应用、非线性薛定谔方程（包括与光子学相关的方程）以及非线性波中的许多其他问题。我的研究主要涉及理论分析，***有助于理解偏微分方程。然而在过去，它对其他学科产生了影响，这些学科的进步有助于更有效的数值模拟或对实验进行更深入的概念理解。我经常与物理科学家合作开展与波动现象有关的领域的项目。 *** *本提案旨在研究物理科学中出现的特定偏微分方程解的性质，这些偏微分方程描述了自由表面水波、非线性光学和连续介质力学等波动现象。这包括色散演化方程（例如非线性薛定谔方程和克莱因-戈登方程）、欧拉方程的涡丝演化以及水波问题。我还计划继续之前对不可压缩纳维-斯托克斯方程的分析。*** ***该程序有三个主要目标： ***(1) 从相空间分析的角度解决哈密顿偏微分方程，将哈密顿系统的精细分析技术（例如 KAM 理论、范式和 Nekhoroshev 稳定性）扩展到自然构成的适当无限维设置中的偏微分方程动力学。***(2) 研究自由表面多种设置中的水波，并使用其哈密顿偏微分方程的性质来描述非线性相互作用、孤立波碰撞和渐近缩放机制，例如使用均质化技术来描述可变测深上的波浪。***(3) 考虑方面纳维-斯托克斯方程的正则理论，包括光谱行为上限的分支，以及奇异集上最近的微局域下界。***我总是让博士后研究员和研究生参与我的研究项目。此外，我打算解决对其他科学领域具有潜在价值的问题，并且我计划继续寻求与非数学家物理科学家的有趣合作，正如我过去成功追求的那样。**