Ebook: Attractors for infinite-dimensional non-autonomous dynamical systems

The book treats the theory of attractors for non-autonomous dynamical systems. The aim of the book is to give a coherent account of the current state of the theory, using the framework of processes to impose the minimum of restrictions on the nature of the non-autonomous dependence.

The book is intended as an up-to-date summary of the field, but much of it will be accessible to beginning graduate students. Clear indications will be given as to which material is fundamental and which is more advanced, so that those new to the area can quickly obtain an overview, while those already involved can pursue the topics we cover more deeply.

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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. Jos? A. Langa is a Profesor Titular at the University of Seville, Spain. James C. Robinson is a Professor at the University of Warwick, UK.

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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. Jos? A. Langa is a Profesor Titular at the University of Seville, Spain. James C. Robinson is a Professor at the University of Warwick, UK.

Content:
Front Matter....Pages i-xxxvi
Front Matter....Pages 1-1
The pullback attractor....Pages 3-22
Existence results for pullback attractors....Pages 23-53
Continuity of attractors....Pages 55-70
Finite-dimensional attractors....Pages 71-102
Gradient semigroups and their dynamical properties....Pages 103-139
Front Matter....Pages 141-141
Semilinear differential equations....Pages 143-186
Exponential dichotomies....Pages 187-222
Hyperbolic solutions and their stable and unstable manifolds....Pages 223-251
Front Matter....Pages 253-253
A non-autonomous competitive Lotka–Volterra system....Pages 255-263
Delay differential equations....Pages 265-279
The Navier–Stokes equations with non-autonomous forcing....Pages 281-300
Applications to parabolic problems....Pages 301-315
A non-autonomous Chafee–Infante equation....Pages 317-338
Perturbation of diffusion and continuity of global attractors with rate of convergence....Pages 339-359
A non-autonomous damped wave equation....Pages 361-376
Appendix: Skew-product flows and the uniform attractor....Pages 377-391
Back Matter....Pages 393-409

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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. Jos? A. Langa is a Profesor Titular at the University of Seville, Spain. James C. Robinson is a Professor at the University of Warwick, UK.

Content:
Front Matter....Pages i-xxxvi
Front Matter....Pages 1-1
The pullback attractor....Pages 3-22
Existence results for pullback attractors....Pages 23-53
Continuity of attractors....Pages 55-70
Finite-dimensional attractors....Pages 71-102
Gradient semigroups and their dynamical properties....Pages 103-139
Front Matter....Pages 141-141
Semilinear differential equations....Pages 143-186
Exponential dichotomies....Pages 187-222
Hyperbolic solutions and their stable and unstable manifolds....Pages 223-251
Front Matter....Pages 253-253
A non-autonomous competitive Lotka–Volterra system....Pages 255-263
Delay differential equations....Pages 265-279
The Navier–Stokes equations with non-autonomous forcing....Pages 281-300
Applications to parabolic problems....Pages 301-315
A non-autonomous Chafee–Infante equation....Pages 317-338
Perturbation of diffusion and continuity of global attractors with rate of convergence....Pages 339-359
A non-autonomous damped wave equation....Pages 361-376
Appendix: Skew-product flows and the uniform attractor....Pages 377-391
Back Matter....Pages 393-409
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