Spectral theory of differential and weighted composition operators

微分和加权合成算子的谱理论

• 批准号: 0354339
• 负责人: Yuri Latushkin
• 金额: $ 11.07万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354339&HistoricalAwards=false
• 关键词: Spectral; theory; differential; weighted; composition

The main theoretical component of this project is to study Fredholm differential operators with unbounded operator coefficients and their Fredholm index in terms of dynamical properties of linear differential equations on Banach spaces. This includes an operator-theoretical treatment of Evans function as a Fredholm determinant. New methods will be developed in spectral analysis of weighted composition (evolution) semigroups and related abstract differential operators on Sobolev and Lebesgue spaces of Banach space-valued functions. Also, we will study connections between the weighted composition operators and associated differential operators, and topics such as maximal regularity of solutions of evolution problems, Atiyah-Patodi-Singer Spectral Flow and Index Theorem, infinite dimensional Morse Theory, Krein-Birman spectral shift function, and Fredholm determinants for operators with semi-separable kernels. The main applied component of this project is to use these methods to study the point and essential spectrum of the classical linearized Euler operator in dimensions two and three obtained by linearizing the Euler equations of hydrodynamics about a steady state.This project lies at the intersection of the theory of complex dynamical systems, the mathematical theory of equations containing many parameters, and stability theory of hydrodynamics describing the behavior of a fluid in equilibria. We will develop new methods of study of the weighted composition semigroups, the mathematical models describing various effects of transport along trajectories of the given complex system. These methods will be applied to the study of a linear approximation of the Euler equations governing the motion of an ideal incompressible fluid in dimensions two and three, which will add to our understanding of fluid dynamics near equilibria.

该项目的主要理论部分是根据 Banach 空间上线性微分方程的动力学性质，研究具有无界算子系数的 Fredholm 微分算子及其 Fredholm 指数。这包括将埃文斯函数作为 Fredholm 行列式的算子理论处理。将开发新方法，用于巴拿赫空间值函数的索博列夫和勒贝格空间上的加权合成（演化）半群和相关抽象微分算子的谱分析。此外，我们还将研究加权合成算子和相关微分算子之间的联系，以及诸如进化问题解的最大正则性、Atiyah-Patodi-Singer 谱流和指数定理、无限维莫尔斯理论、Krein-Birman 谱位移函数等主题，以及具有半可分离核的算子的 Fredholm 行列式。该项目的主要应用部分是利用这些方法来研究通过对稳态流体动力学欧拉方程进行线性化而获得的经典线性化欧拉算子在二维和三维上的点和本质谱。复杂动力系统理论、包含许多参数的方程的数学理论以及描述平衡状态下流体行为的流体动力学稳定性理论。我们将开发研究加权复合半群的新方法，即描述沿给定复杂系统轨迹传输的各种影响的数学模型。这些方法将应用于研究控制二维和三维理想不可压缩流体运动的欧拉方程的线性近似，这将增加我们对接近平衡状态的流体动力学的理解。