Spectral analysis of the Koopman operator for partial differential equations
Abstract
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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Nov 01, 2020
- Source ID
- 10.1063/5.0011470
Entities
People
- Hiroya Nakao
- Igor Mezić
Organizations
- Core Research for Evolutional Science and Technology
- Japan Society for the Promotion of Science
- Tokyo Institute of Technology