Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems

ISBN-13: 9781489991768 / Angielski / Miękka / 2014 / 412 str.

Carvalho, Alexandre

Opis

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.

Recenzje

From the reviews:

"The Carvalho, Langa and Robinson monograph focuses primarily on infinite-dimensional systems and evolution equations. ... The monograph is suitable for graduate students with a background on functional analysis and evolution equations. ... Carvalho, Langa and Robinson present a readable and thorough account of the current state of the theory, which provides the reader with ready access to an area that has considerable potential for further development." (Peter E. Kloeden, Mathematical Reviews, July, 2013)

"This monograph not only summarizes the research of the authors over the last decade, but also provides an accessible and well-written approach to the recent theory of non-autonomous dynamical systems in infinite dimensions with a focus on corresponding attractors and invariant manifolds. ... This book is a well-written and carefully prepared text appropriate for advanced classes on dynamical systems and seminars." (Christian Pötzsche, Zentralblatt MATH, Vol. 1263, 2013)

Spis treści

The pullback attractor.- Existence results for pullback attractors.- Continuity of attractors.- Finite-dimensional attractors.- Gradient semigroups and their dynamical properties.- Semilinear Differential Equations.- Exponential dichotomies.- Hyperbolic solutions and their stable and unstable manifolds.- A non-autonomous competitive Lotka-Volterra system.- Delay differential equations.-The Navier–Stokes equations with non-autonomous forcing.-  Applications to parabolic problems.- A non-autonomous Chafee–Infante equation.- Perturbation of diffusion and continuity of attractors with rate.- A non-autonomous damped wave equation.- References.- Index.-

O autorze

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

Blurb

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.