Research in Analysis

分析研究

• 批准号: 1464479
• 负责人: Sergey Denisov
• 金额: $ 21.81万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464479&HistoricalAwards=false
• 关键词: Research; Analysis

This research project covers a range of problems central to classical analysis and partial differential equations. The research will focus on three topics: approximation theory, nonlinear nonlocal evolution equations, and the spectral theory of Schrodinger operators. In approximation theory, the research concerns problems related to estimating the size of orthonormal polynomials. Understanding the asymptotics of the orthonormal polynomials has proved to be crucial for the progress in several fields, including mathematical physics and probability. The second main theme of the research is analysis of two-dimensional active scalar equations. These nonlinear and nonlocal evolution equations appear in the fluid dynamics and atmospheric and ocean sciences. The physical processes described by these equations are known to be highly unstable, and this project aims to develop a mathematical explanation for that phenomenon. The third topic of this project is spectral theory of Schrodinger operators. The Schrodinger equation is one of the basic equations in mathematical physics. The project will focus on understanding how the properties of the potential function affect the scattering of waves. In the area of approximation theory, the research investigates the following classical question: what is the size of the polynomial orthonormal on the unit circle (or the real line) with respect to a measure from a given class. Various ways to define the size and various classes of measures will be considered. For example, the project will study a class of weights that have a fixed oscillation around a given constant. The second part of the project will mainly focus on two active scalar equations: the two-dimensional Euler equation of fluid dynamics and the surface quasi-geostrophic equation. Questions of instability will be addressed; in particular, patch dynamics will be studied and the merging mechanism of a centrally symmetric pair will be analyzed. The third topic of this project will address multidimensional scattering. The Schrodinger evolution with rough potential will be considered and the classical questions of the scattering theory will be studied. To carry out the proposed research, methods of harmonic and complex analysis will be used along with techniques from nonlinear analysis and the theory of partial differential equations. Through the training of graduate and undergraduate students the project will have an impact on human resource development.

这个研究项目涵盖了一系列经典分析和偏微分方程式的核心问题。研究将集中在三个主题上：逼近理论、非线性非局域发展方程和薛定谔算子的谱理论。在逼近理论中，研究了与估计正交多项式的大小有关的问题。了解正交化多项式的渐近性对于数学物理和概率等多个领域的发展是至关重要的。研究的第二个主要主题是二维活动标量方程的分析。这些非线性的、非局部的演化方程出现在流体力学、大气和海洋科学中。众所周知，这些方程所描述的物理过程是高度不稳定的，这个项目的目的是为这一现象发展一个数学解释。本项目的第三个主题是薛定谔算子的谱理论。薛定谔方程是数学物理中的基本方程之一。该项目将侧重于了解势函数的性质如何影响波的散射。在逼近理论领域，本研究研究了以下经典问题：单位圆(或实直线)上的多项式正交线相对于某一给定类的度量的大小是多少。将考虑以各种方式确定措施的规模和各类措施。例如，该项目将研究一类围绕给定常量有固定振荡的权重。该项目的第二部分将主要关注两个活跃的标量方程：二维欧拉流体动力学方程和地表准地转方程。本课程将讨论不稳定性问题，特别是将研究补丁动力学，并分析中心对称对的合并机制。本项目的第三个主题将解决多维散射问题。我们将考虑粗糙势下的薛定谔演化，并研究散射理论的经典问题。为了开展所提出的研究，将使用调和和复变分析方法，以及非线性分析和偏微分方程组理论。通过对研究生和本科生的培训，该项目将对人力资源开发产生影响。

项目成果

On the Growth of the Support of Positive Vorticity for 2D Euler Equation in an Infinite Cylinder

无限圆柱内二维欧拉方程正涡度支持度的增长

• DOI: 10.1007/s00220-019-03295-w
• 发表时间: 2019
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Choi, Kyudong;Denisov, Sergey
• 通讯作者: Denisov, Sergey

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

树和多个正交多项式上的自伴雅可比矩阵

• DOI: 10.1090/tran/7959
• 发表时间: 2020
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Aptekarev, Alexander I.;Denisov, Sergey A.;Yattselev, Maxim L.
• 通讯作者: Yattselev, Maxim L.