Attractors of Evolution Equations

Description

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. 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