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**Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors.**
*(English)*
Zbl 0980.35001

Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press. xvii, 461 p. (2001).

This 461 pages long monograph brings a nice exposition of the theory of dynamical systems governed by partial differential equations. Intended a ‘didactic text’ for graduate students it develops the core of the theory from its foundations up to the recent concepts crucial in current research.

The monograph consists of four parts, a description of which is as follows.

Part I contains Chapters 1-5 with introductory material summarizing basic topics from functional analysis (Banach and Hilbert spaces), ordinary differential equations (well-posedness result), linear operators (Hilbert-Schmidt type theorem), dual spaces (weak convergence and weak compactness), and Sobolev spaces (strongly emphasizing Hilbert \(H^s(\Omega)\) spaces).

In four next chapters of Part II several topics from the theory of partial differential equations are discussed. Chapter 6 brings the results concerning solutions of elliptic problems involving Laplacian with periodic or Dirichlet boundary conditions. Chapter 7 deals with a linear evolutionary equation in a Hilbert space and describes the Galerkin method in great detail. The approach based on Galerkin approximations is next used in the studies of semilinear parabolic equations. Namely, semigroups of global solutions corresponding to a scalar reaction-diffusion equation and to 2D Navier-Stokes system are constructed in the successive Chapters 8 and 9. These two equations are chosen as representative applications and accompany the reader in a large portion of the book.

Part III expounds the theory of global attractors (in Chapter 10) and gives the construction of a global attractor in case of two representative problems mentioned above (in Chapters 11 and 12). In Chapter 13 an important topic concerning finite dimension of attractor is reviewed and examples of bounds on the attractor dimension are given.

Part IV is mostly devoted to the description of finite-dimensional dynamics. In Chapter 14 the squeezing property is introduced together with the concepts of inertial manifolds and exponential attractors. A squeezing property reappears in Chapter 15 in a stronger form suitable for the considerations of inertial manifolds. Chapter 16 presents the idea of constructing a finite-dimensional system ‘reproducing’ the dynamics on a global attractor. The final Chapter 17 provides an analysis of 1D Kuramoto-Sivashinsky equation via a number of exercises. Exercises and comments concerning the bibliography enrich all chapters of the monograph.

The book is written as a nice introduction to the – probably most complete nowadays monograph devoted to the theory of global attractors – Infinite-Dimensional Dynamical Systems in Mechanics and Physics by R. Temam and will undoubtedly be helpful for graduate students in their preliminary study of this subject. However, the book does not give a wider review of the recent approach to solvability of parabolic equations (related to D. Henry [Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840 (1981; Zbl 0456.35001)]); the monographs by A. Pazy [Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, New York etc.: Springer-Verlag (1983; Zbl 0516.47023)], H. Amann [Linear and quasilinear parabolic problems, Vol. 1: Abstract linear theory, Monographs in Mathematics, 89, Basel: Birkhäuser (1995; Zbl 0819.35001)], A. Lunardi [Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, 16, Basel: Birkhäuser (1995; Zbl 0816.35001)] do not appear even in the references. Such an approach may also be important for students since it allows for the unified treatment of the existence of global attractors for a large class of parabolic equations and systems.

A particular intention of the author – to produce the sort of text a student would like to read – came true. The book meets these expectations and will certainly benefit young researchers entering the described field.

The monograph consists of four parts, a description of which is as follows.

Part I contains Chapters 1-5 with introductory material summarizing basic topics from functional analysis (Banach and Hilbert spaces), ordinary differential equations (well-posedness result), linear operators (Hilbert-Schmidt type theorem), dual spaces (weak convergence and weak compactness), and Sobolev spaces (strongly emphasizing Hilbert \(H^s(\Omega)\) spaces).

In four next chapters of Part II several topics from the theory of partial differential equations are discussed. Chapter 6 brings the results concerning solutions of elliptic problems involving Laplacian with periodic or Dirichlet boundary conditions. Chapter 7 deals with a linear evolutionary equation in a Hilbert space and describes the Galerkin method in great detail. The approach based on Galerkin approximations is next used in the studies of semilinear parabolic equations. Namely, semigroups of global solutions corresponding to a scalar reaction-diffusion equation and to 2D Navier-Stokes system are constructed in the successive Chapters 8 and 9. These two equations are chosen as representative applications and accompany the reader in a large portion of the book.

Part III expounds the theory of global attractors (in Chapter 10) and gives the construction of a global attractor in case of two representative problems mentioned above (in Chapters 11 and 12). In Chapter 13 an important topic concerning finite dimension of attractor is reviewed and examples of bounds on the attractor dimension are given.

Part IV is mostly devoted to the description of finite-dimensional dynamics. In Chapter 14 the squeezing property is introduced together with the concepts of inertial manifolds and exponential attractors. A squeezing property reappears in Chapter 15 in a stronger form suitable for the considerations of inertial manifolds. Chapter 16 presents the idea of constructing a finite-dimensional system ‘reproducing’ the dynamics on a global attractor. The final Chapter 17 provides an analysis of 1D Kuramoto-Sivashinsky equation via a number of exercises. Exercises and comments concerning the bibliography enrich all chapters of the monograph.

The book is written as a nice introduction to the – probably most complete nowadays monograph devoted to the theory of global attractors – Infinite-Dimensional Dynamical Systems in Mechanics and Physics by R. Temam and will undoubtedly be helpful for graduate students in their preliminary study of this subject. However, the book does not give a wider review of the recent approach to solvability of parabolic equations (related to D. Henry [Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840 (1981; Zbl 0456.35001)]); the monographs by A. Pazy [Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, New York etc.: Springer-Verlag (1983; Zbl 0516.47023)], H. Amann [Linear and quasilinear parabolic problems, Vol. 1: Abstract linear theory, Monographs in Mathematics, 89, Basel: Birkhäuser (1995; Zbl 0819.35001)], A. Lunardi [Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, 16, Basel: Birkhäuser (1995; Zbl 0816.35001)] do not appear even in the references. Such an approach may also be important for students since it allows for the unified treatment of the existence of global attractors for a large class of parabolic equations and systems.

A particular intention of the author – to produce the sort of text a student would like to read – came true. The book meets these expectations and will certainly benefit young researchers entering the described field.

Reviewer: Jan Cholewa (Katowice)