Exponential attractors for dissipative evolution equations.

*(English)*Zbl 0842.58056
Research in Applied Mathematics. 37. Chichester: Wiley. Paris: Masson. viii, 182 p. (1994).

The book under review is devoted to a new direction of the theory of attractors for dissipative evolution equations. Exponential attractors (EA) are the objects intermediate between global attractors and inertial manifolds. The introduction contains the history of this subject. The following four chapters give the notion of EAs and their construction for dissipative evolution equations of the first order, approximations of EAs, in particular Galerkin approximations. The chapter 5 contains the applications to Kuramoto-Sivashinsky, Kuramoto-Sivashinsky-Spiegel equations, 2D and 3D Navier-Stokes equations, Burgers equations and Chaffee-Infante reaction-diffusion equations. In the chapter 6 this theory is developed for second order evolution equations with damping. The applications to sine-Gordon, Klein-Gordon type equations, to systems of sine-Gordon equations are given. The chapter 7 is devoted to EAs of optimal Hausdorff dimension and optimal outer Lyapunov dimension. The concluding chapters 9-10 contain a review about inertial manifolds and their comparison with EAs, Mañé’s projections theory and inertially equivalent dynamical systems. Necessary additional material about Mañé’s theorem for Hilbert spaces, estimate of the topological entropy and mathematical background of fractal sets are carried out into appendices A, B and C. Thus the purpose of this work is to develop and present the theory of exponential attractors for dissipative evolution equations of infinite dimension. This book can be considered for a graduate course. It is very interesting because of pointing out various connections between fluid mechanics, partial differential equations and dynamical systems.

Reviewer: B.V.Loginov (Ul’yanovsk)

##### MSC:

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

35Q30 | Navier-Stokes equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |