Spectral analysis of the Koopman operator for partial differential equations

Area of Science:

• Dynamical Systems Theory
• Mathematical Physics
• Nonlinear Dynamics

Background:

• Partial differential equations (PDEs) model systems relaxing to stable states.
• Understanding the long-term behavior of these systems is crucial.
• Existing methods may not fully capture the spectral properties of the dynamics.

Purpose of the Study:

• To introduce Koopman-operator analysis for PDEs describing relaxation dynamics.
• To develop spectral expansions of the Koopman operator using Koopman eigenfunctions.
• To connect Koopman analysis concepts to established dynamical systems notions like inertial manifolds and isostables.

Main Methods:

• Koopman-operator theory applied to PDEs.
• Introduction and utilization of Koopman eigenfunctions.
• Spectral expansion of the Koopman operator via conjugacy.
• Analysis of linear and nonlinear PDEs as examples.

Main Results:

• Koopman eigenfunctions are introduced for relaxation dynamics.
• Spectral expansion of the Koopman operator is derived.
• Koopman eigenfunctions relate to linear functionals for linear systems.
• Inertial manifolds correspond to zero level sets of Koopman eigenfunctions.
• Isostables are defined by the slowest decaying Koopman eigenfunctional.

Conclusions:

• Koopman-operator analysis offers a powerful spectral framework for PDEs.
• The framework provides new perspectives on inertial manifolds and isostables.
• The method is demonstrated on diffusion, Burgers, and phase-diffusion equations.