Attractors of Evolution Equations

A.V. Babin and M.I. Vishik (Eds.)0444890041, 9780444890047, 9780080875460

Problems, ideas and notions from the theory of finite-dimensional dynamical systems have penetrated deeply into the theory of infinite-dimensional systems and partial differential equations. From the standpoint of the theory of the dynamical systems, many scientists have investigated the evolutionary equations of mathematical physics. Such equations include the Navier-Stokes system, magneto-hydrodynamics equations, reaction-diffusion equations, and damped semilinear wave equations. Due to the recent efforts of many mathematicians, it has been established that the attractor of the Navier-Stokes system, which attracts (in an appropriate functional space) as t – # all trajectories of this system, is a compact finite-dimensional (in the sense of Hausdorff) set. Upper and lower bounds (in terms of the Reynolds number) for the dimension of the attractor were found. These results for the Navier-Stokes system have stimulated investigations of attractors of other equations of mathematical physics.

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Table of contents :
Content:
Edited by
Pages ii-iii

Copyright page
Page iv

Preface
Pages v-vi
A. Babin

Introduction
Pages 1-12

Chapter 1 Quasilinear Evolutionary Equations and Semigroups Generated by Them
Pages 13-116

Chapter 2 Maximal Attractors of Semigroups
Pages 117-155

Chapter 3 Attractors and Unstable Sets
Pages 157-191

Chapter 4 Some Information on Semigroups of Linear Operators
Pages 193-216

Chapter 5 Invariant Manifolds of Semigroups and Mappings at Equilibrium Points
Pages 217-296

Chapter 6 Steady-State Solutions
Pages 297-340

Chapter 7 Differentiability of Operators of Semigroups Generated by Partial Differential Equations
Pages 341-398

Chapter 8 Semigroups Depending on a Parameter
Pages 399-422

Chapter 9 Dependence on a Parameter of Attractors of Differentiate Semigroups and Uniform Asymptotics of Trajectories
Pages 423-477

Chapter 10 Hausdorff Dimension of Attractors
Pages 479-503

Bibliography Review Article
Pages 505-526

Index
Pages 527-532