偏微分方程Koopman算子的谱分析
Spectral Analysis of the Koopman Operator for Partial Differential Equations
摘要：
我们提供了一个概述的Koopman算子分析一类偏微分方程描述松弛的字段变量到一个稳定的定态。我们引入Koopman本征泛函的系统，并使用共轭的概念来开发频谱扩展的Koopman运营商。对于线性系统，如扩散方程，库普曼本征泛函可以表示为场变量的线性泛函。惯性流形的概念对应于库普曼本征泛函的联合零水平集，而等稳的概念被定义为最慢衰减的库普曼本征泛函的水平集。以线性扩散方程、非线性Burgers方程和非线性相扩散方程为例进行了分析。
英文摘要：
We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the notion of conjugacy to develop spectral expansion of the Koopman operator. For linear systems such as the diffusion equation, the Koopman eigenfunctionals can be expressed as linear functionals of the field variable. The notion of inertial manifolds is shown to correspond to joint zero level sets of Koopman eigenfunctionals, and the notion of isostables is defined as the level sets of the slowest decaying Koopman eigenfunctional. Linear diffusion equation, nonlinear Burgers equation, and nonlinear phase-diffusion equation are analyzed as examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
