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.

该研究项目涵盖了经典分析和部分微分方程中心的一系列问题。该研究将重点介绍三个主题：近似理论，非线性非局部进化方程和施罗宾格运营商的光谱理论。在近似理论中，该研究涉及与估计正统多项式大小有关的问题。事实证明，了解正规多项式的渐近学对于多个领域的进步至关重要，包括数学物理学和概率。该研究的第二个主题是对二维活动标量方程的分析。这些非线性和非局部进化方程出现在流体动力学以及大气和海洋科学中。这些方程式描述的物理过程已知高度不稳定，该项目旨在为该现象开发数学解释。该项目的第三个主题是Schrodinger操作员的光谱理论。 Schrodinger方程是数学物理学的基本方程之一。该项目将集中于了解电势函数的性质如何影响波的散射。在近似理论的领域中，研究研究了以下经典问题：关于单位圆（或实际线）在给定类别的度量方面多项式正交的大小是多少。将考虑定义大小和各种措施的各种方法。例如，该项目将研究一类权重，这些权重围绕给定常数进行固定振荡。 该项目的第二部分将主要集中在两个主动标量方程上：流体动力学的二维Euler方程和表面准地藻方程。不稳定问题将被解决；特别是，将研究斑块动力学，并将分析中央对称对的合并机制。该项目的第三个主题将解决多维散射。将考虑具有粗略潜力的施罗辛格进化，并将研究散射理论的经典问题。为了进行拟议的研究，将使用谐波和复杂分析的方法与非线性分析和部分微分方程理论一起使用。通过培训研究生和本科生，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.