Analysis of evolution equations and related problems

演化方程及相关问题分析

• 批准号: 1067413
• 负责人: Sergey Denisov
• 金额: $ 17.07万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1067413&HistoricalAwards=false
• 关键词: Analysis; evolution; equations; related; problems

The main goal of this project is to study the long-time behavior of certain linear and nonlinear evolution equations. In the linear case, the problems of interest include the following: the long-time behavior and growth of Sobolev norms of solutions to evolution equations when the value of the coupling constant is generic; applications to multidimensional scattering for Schrodinger operators with slowly decaying or random potentials; the analysis of the spatial asymptotics of Green's function for Schrodinger equations; and sharp results for the dynamics in one-dimensional wave equations with decaying potentials. The nonlinear evolution equations to be studied in this project have their origins in two-dimensional incompressible fluid dynamics. Specifically, the equations to be studied are the Euler equation and the surface quasigeostrophic equation. The project will address the issue of instability for these equations. In the case of the Euler equation, the problem of optimal growth of the Sobolev norms of vorticity will be studied, while in the quasigeostrophic setting the scenario of blow-up in finite time will be considered. To make a progress, the principal investigator will use the tools of harmonic analysis (multilinear operators, potential theory, harmonic measure, singular integrals), approximation theory (polynomials orthogonal on the circle and on the real line), probability (Ito's calculus), and spectral theory for self-adjoint operators and hyperbolic pencils. This project will focus on mathematical problems that are central to quantum and fluid mechanics. Quantum mechanics, which was created and developed in the last century, is a basic branch of the modern physics, and the dynamics of fluids is another branch of physics studied as early as the eighteenth century by Leonhard Euler. The analysis of evolution equations and wave propagation in the presence of rough or random medium suggested in this project is a central problem of quantum mechanics, so this research has a potential impact on the development of that field. One of the most intriguing problems in fluid dynamics is the problem of singularity formation. This phenomenon is ubiquitous in nature and one goal of this project is to study its mechanism mathematically by focusing on some simplified two-dimensional models. To accomplish these goals, tools from various areas of mathematics will be refined and applied, which will advance these fields as well. The work on the project will include mentoring graduate students and coaching undergraduate research teams. This will have an additional impact on human resource development.

本课题的主要目的是研究某些线性和非线性演化方程的长期行为。在线性情况下，关注的问题包括：当耦合常数为一般值时，演化方程解的Sobolev范数的长期行为和增长；具有慢衰减或随机势的薛定谔算符在多维散射中的应用薛定谔方程格林函数的空间渐近性分析；并对具有衰减势的一维波动方程的动力学得到了清晰的结果。本课题研究的非线性演化方程来源于二维不可压缩流体动力学。具体来说，要研究的方程是欧拉方程和曲面拟地转方程。该项目将解决这些方程的不稳定性问题。对于欧拉方程，将研究涡度Sobolev范数的最优增长问题，而在准等转情况下，将考虑有限时间内爆炸的情况。为了取得进展，首席研究员将使用调和分析（多线性算子、势理论、调和测度、奇异积分）、近似理论（圆上和实线上正交的多项式）、概率（伊藤微积分）和自伴随算子和双曲铅笔的谱理论等工具。这个项目将集中于量子和流体力学的核心数学问题。量子力学是上个世纪创立和发展起来的，是现代物理学的一个基本分支，而流体动力学是物理学的另一个分支，早在18世纪就由莱昂哈德·欧拉研究过。本课题提出的粗糙或随机介质下的演化方程和波传播分析是量子力学的核心问题，因此本研究对量子力学的发展具有潜在的影响。流体动力学中最有趣的问题之一是奇点的形成问题。这种现象在自然界是普遍存在的，本项目的目标之一是通过一些简化的二维模型来研究其数学机制。为了实现这些目标，来自各个数学领域的工具将被改进和应用，这也将推动这些领域的发展。该项目的工作将包括指导研究生和指导本科生研究团队。这将对人力资源发展产生额外的影响。