  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.
