Attractors of Evolution Equations


  • A.V. Babin, Moscow Institute for Railroad, Transportation Engineers (MIIT), Moscow, Russia
  • M.I. Vishik, Moscow State University, Moscow, Russia

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. For certain problems, in particular for reaction-diffusion systems and nonlinear damped wave equations, mathematicians have established the existence of the attractors and their basic properties; furthermore, they proved that, as t - +∞, an infinite-dimensional dynamics described by these equations and systems uniformly approaches a finite-dimensional dynamics on the attractor U, which, in the case being considered, is the union of smooth manifolds. This book is devoted to these and several other topics related to the behaviour as t - ∞ of solutions for evolutionary equations.
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Book information

  • Published: March 1992
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-89004-7

Reviews excellent introduction to a difficult subject.
Mathematical Reviews

Table of Contents

Quasilinear Evolutionary Equations and Semigroups Generated by Them. Maximal Attractors of Semigroups. Attractors and Unstable Sets. Some Information on Semigroups of Linear Operators. Invariant Manifolds of Semigroups and Mapping at Equilibrium Points. Steady-state Solutions. Differentiability of Operators of Semigroups Generated by Partial Differential Equations. Semigroups Depending on a Parameter. Dependence on a Parameter of Attractors of Differentiable Semigroups and Uniform Asymptotics of Trajectories. Hausdorff Dimension of Attractors. Bibliography. Index.