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Attractors for infinite-dimensional non-autonomous dynamical systems

  • Alexandre N. Carvalho
  • José A. Langa
  • James C. Robinson

Part of the Applied Mathematical Sciences book series (AMS, volume 182)

Table of contents

  1. Front Matter
    Pages i-xxxvi
  2. Abstract theory

    1. Front Matter
      Pages 1-1
    2. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 3-22
    3. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 23-53
    4. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 55-70
    5. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 71-102
    6. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 103-139
  3. Invariant manifolds of hyperbolic solutions

    1. Front Matter
      Pages 141-141
    2. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 143-186
    3. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 187-222
    4. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 223-251
  4. Applications

    1. Front Matter
      Pages 253-253
    2. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 255-263
    3. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 265-279
    4. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 281-300
    5. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 301-315
    6. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 317-338
    7. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 339-359
    8. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 361-376
    9. Alexandre N. Carvalho, José A. Langa, James C. Robinson
      Pages 377-391
  5. Back Matter
    Pages 393-409

About this book

Introduction

This book treats the theory of pullback attractors for non-autonomous dynamical systems. While the emphasis is on infinite-dimensional systems, the results are also applied to a variety of finite-dimensional examples.

 

The purpose of the book is to provide a summary of the current theory, starting with basic definitions and proceeding all the way to state-of-the-art results. As such it is intended as a primer for graduate students, and a reference for more established researchers in the field.

 

The basic topics are existence results for pullback attractors, their continuity under perturbation, techniques for showing that their fibres are finite-dimensional, and structural results for pullback attractors for small non-autonomous perturbations of gradient systems (those with a Lyapunov function).  The structural results stem from a dynamical characterisation of autonomous gradient systems, which shows in particular that such systems are stable under perturbation.

Application of the structural results relies on the continuity of unstable manifolds under perturbation, which in turn is based on the robustness of exponential dichotomies: a self-contained development of  these topics is given in full.

After providing all the necessary theory the book treats a number of model problems in detail, demonstrating the wide applicability of the definitions and techniques introduced: these include a simple Lotka-Volterra ordinary differential equation, delay differential equations, the two-dimensional Navier-Stokes equations, general reaction-diffusion problems, a non-autonomous version of the Chafee-Infante problem, a comparison of attractors in problems with perturbations to the diffusion term, and a non-autonomous damped wave equation.

Alexandre N. Carvalho is a Professor at the University of Sao Paulo, Brazil. José A. Langa is a Profesor Titular at the University of Seville, Spain. James C. Robinson is a Professor at the University of Warwick, UK.

Keywords

Autonomous systems Dynamical Systems Navier-Stokes equations non-autonomous theory

Authors and affiliations

  • Alexandre N. Carvalho
    • 1
  • José A. Langa
    • 2
  • James C. Robinson
    • 3
  1. 1.Universidade de São PauloInstituto de Ciências Matemáticas e de CSão Carlos SPBrazil
  2. 2., Dpto. EDANUniversidad de SevillaSevilleSpain
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-4581-4
  • Copyright Information Springer Science+Business Media, LLC 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-4580-7
  • Online ISBN 978-1-4614-4581-4
  • Series Print ISSN 0066-5452
  • Buy this book on publisher's site