James C. Robinson.
Infinite-Dimensional Dynamical Systems: an introduction to dissipative Parabolic PDEs and the theory of global attractors.
CUP. 2001
This book develops the theory of global attractors for a class of parabolic PDEs
that includes reaction—diffusion equations and the Navier-Stokes equations,
two examples that are treated in detail.
A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment
of existence and uniqueness of solutions for both linear time-independent problems
(Poisson’s equation) and the nonlinear evolution equations that generate
the infinite-dimensional dynamical systems of the title.
Attention then turns to the global attractor,
a finite-dimensional subset of the infinite-dimensional phase space
that determines the asymptotic dynamics.
In particular, the concluding chapters investigate in what sense
the dynamics restricted to the attractor are themselves “finite-dimensional.”
The book is intended as a didactic text for first-year graduate students and assumes only a basic knowledge of elementary functional analysis.