Analysis of evolution equations and related problems

这个项目的主要目的是研究某些线性和非线性发展方程的长时间行为。在线性情形下，我们感兴趣的问题包括：当耦合常数的值是一般的时，发展方程解的Soblev范数的长时间行为和增长；具有缓慢衰减势或随机势的薛定谔算子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；具有衰减势的一维波动方程动力学的精确结果。本项目研究的非线性发展方程起源于二维不可压缩流体动力学。具体地说，要研究的方程是欧拉方程和地表准地转方程。该项目将解决这些方程的不稳定性问题。在欧拉方程的情况下，将研究涡度的Sobolev范数的最优增长问题，而在准地转的情况下，将考虑有限时间内的爆破情形。为了取得进展，主要研究人员将使用调和分析(多线性算子、位势理论、调和测量、奇异积分)、逼近理论(圆周上和实线上正交的多项式)、概率(Ito演算)和自伴算子和双曲铅笔的谱理论等工具。这个项目将专注于量子和流体力学的核心数学问题。上个世纪产生和发展起来的量子力学是现代物理学的一个基本分支，而流体动力学是18世纪欧拉研究的另一个物理学分支。本项目提出的粗糙或随机介质中演化方程和波传播的分析是量子力学的中心问题，因此这一研究对该领域的发展具有潜在的影响。流体力学中最耐人寻味的问题之一是奇点的形成问题。这种现象在自然界中是普遍存在的，本项目的目标之一是通过一些简化的二维模型对其机理进行数学研究。为了实现这些目标，来自数学各个领域的工具将得到改进和应用，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。这将对人力资源开发产生额外影响。

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.