Integral manifolds and inertial manifolds for dissipative partial differential equations. (English) Zbl 0683.58002

Applied Mathematical Sciences, 70. New York etc.: Springer-Verlag. x, 122 p. DM 68.00 (1989).
This book focuses a new geometric construction of inertial manifolds for a class of dissipative partial differential equations. In the Chapters 2 to 14 a general method, based on Cauchy integral manifolds, of constructing these inertial manifolds, is presented. The key geometric property is a spectral blocking property, the method is explicitly constructive and avoiding any fixed point theorem. The Chapters 15 to 19 are devoted to the application of the theory presented in the Chapters 2 to 14. For the reviewer this is the most interesting part of the book. The high flexibility of their method is demonstrated by constructing inertial manifolds for several specific examples. So the reader will find in Chapter 15 the application of the developed theory to the Kuramoto-Sivashinsky equation, in Chapter 16 to the nonlocal Burger equation and in Chapter 17 to the Cahn-Hilliard equation. The remaining two Chapters are devoted to reaction-diffusion equations. So in Chapter 18 a parabolic equation in two space variables is considered and in Chapter 19 the Chaffee-Infante equation. This part of the book shows very impressive how the theory presented in the Chapters 2 to 10 have to be adjusted for the latter applications. This gives a good impression about the presented method which can be readily adapted to the special structure of each of the presented dissipative equations.
This book should become a standard lecture for all who are interested in the field of dissipative partial differential equations.
Reviewer: K.Doppel


37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
58D25 Equations in function spaces; evolution equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
58J99 Partial differential equations on manifolds; differential operators
35K57 Reaction-diffusion equations