chueshov. introduction to infinite-dimensional dynamical and dissipative systems(419s)

I.D.Chueshov D issipative S ystems Infinite-Dimensional Introduction Theory I. D. Chueshov Introduction to the Theory of InfiniteDimensional Dissipative Systems 966–7021–64–5 ORDER www.acta.com.ua I . D . Chueshov Chueshov Universitylecturesincontemporarymathematics D issipative issipative S ystems ystems of Infinite-Dimensional Infinite-Dimensional Introduction ntroduction Theory Theory to the This book provides an exhau - stive introduction to the scope of main ideas and methods of the theory of infinite-dimensional dis - sipative dynamical systems which has been rapidly developing in re - cent years. In the examples sys tems generated by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-dimensional dissipative systems and to qualify him/her for an independent scien - tific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book. The core of the book is com - posed of the courses given by the author at the Department of Me chanics and Mathematics at Kharkov University during a number of years. This book con - tains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book. Translated by Constantin I. Chueshov from the Russian edition («ACTA», 1999) Translation edited by Maryna B. Khorolska Author: I. D. Chueshov I. D. Chueshov Title: Introduction to the Theory Introduction to the Theory of InfiniteDimensional of InfiniteDimensional Dissipative Systems Dissipative Systems ISBN: 966–7021–64–5 You can ORDER ORDER this book while visiting the website of «A CTA» Scientific Publishing House http://www.acta.com.ua www.acta.com.ua /en/ « A CTA » 2002 I. D. Chueshov Introduction IntroductionIntroduction Introduction to the Theory of Infinite-Dimensional to the Theory of Infinite-Dimensionalto the Theory of Infinite-Dimensional to the Theory of Infinite-Dimensional Dissipative Systems Dissipative SystemsDissipative Systems Dissipative Systems A CTA 2002 UDC 517 2000 Mathematics Subject Classification: primary 37L05; secondary 37L30, 37L25. This book provides an exhaustive introduction to the scope of main ideas and methods of the theory of infinite-dimen- sional dissipative dynamical systems which has been rapidly developing in recent years. In the examples systems genera- ted by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are con- sidered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-di- mensional dissipative systems and to qualify him/her for an independent scientific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book. The core of the book is composed of the courses given by the author at the Department of Mechanics and Mathematics at Kharkov University during a number of years. This book contains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book. Translated by Constantin I. Chueshov from the Russian edition ( « A CTA » , 1999) Translation edited by Maryna B. Khorolska A CTA Scientific Publishing House Kharkiv, Ukraine E-mail: we@acta.com.ua © I. D. Chueshov, 1999, 2002 © Series, « A CTA » , 1999 © Typography, layout, « A CTA » , 2002 ISBN 966-7021-20-3 (series) ISBN 966-7021-64-5 Свідоцтво ДК №179 www .acta.com.ua Contents ContentsContents Contents . . . . Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 1 . Basic Concepts of Basic Concepts of Basic Concepts of Basic Concepts of th thth the Theory e Theorye Theory e Theory of Infinite-Dimensional Dynamical Sy of Infinite-Dimensional Dynamical Syof Infinite-Dimensional Dynamical Sy of Infinite-Dimensional Dynamical Syst stst stems emsems ems . . . . § 1 Notion of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . § 2 Trajectories and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . § 3 Definition of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . § 4 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . . 24 . . . . § 5 Theorems on Existence of Global Attractor . . . . . . . . . . . . . . 28 . . . . § 6 On the Structure of Global Attractor . . . . . . . . . . . . . . . . . . . 34 . . . . § 7 Stability Properties of Attractor and Reduction Principle . . 45 . . . . § 8 Finite Dimensionality of Invariant Sets . . . . . . . . . . . . . . . . . 52 . . . . § 9 Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems . . . . . . . . . . . . . 61 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter 2 . Long-Time Behaviour of Solutions Long-Time Behaviour of SolutionsLong-Time Behaviour of Solutions Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations to a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equations to a Class of Semilinear Parabolic Equations . . . . § 1 Positive Operators with Discrete Spectrum . . . . . . . . . . . . . . 77 . . . . § 2 Semilinear Parabolic Equations in Hilbert Space . . . . . . . . . . 85 . . . . § 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 . . . . § 4 Existence Conditions and Properties of Global Attractor . . 101 . . . . § 5 Systems with Lyapunov Function . . . . . . . . . . . . . . . . . . . . . 108 . . . . § 6 Explicitly Solvable Model of Nonlinear Diffusion . . . . . . . . . 118 . . . . § 7 Simplified Model of Appearance of Turbulence in Fluid . . . 130 . . . . § 8 On Retarded Semilinear Parabolic Equations . . . . . . . . . . . 138 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4 Contents Chapter 3 . Inertial Manifolds Inertial ManifoldsInertial Manifolds Inertial Manifolds . . . . § 1 Basic Equation and Concept of Inertial Manifold . . . . . . . . 149 . . . . § 2 Integral Equation for Determination of Inertial Manifold . . 155 . . . . § 3 Existence and Properties of Inertial Manifolds . . . . . . . . . . 161 . . . . § 4 Continuous Dependence of Inertial Manifold on Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 . . . . § 5 Examples and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 . . . . § 6 Approximate Inertial Manifolds for Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . 182 . . . . § 7 Inertial Manifold for Second Order in Time Equations . . . . 189 . . . . § 8 Approximate Inertial Manifolds for Second Order in Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 . . . . § 9 Idea of Nonlinear Galerkin Method . . . . . . . . . . . . . . . . . . . 209 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Chapter 4 . The Problem on Nonlinear The Problem on NonlinearThe Problem on Nonlinear The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow Oscillations of a Plate in a Supersonic Gas FlowOscillations of a Plate in a Supersonic Gas Flow Oscillations of a Plate in a Supersonic Gas Flow . . . . § 1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 . . . . § 2 Auxiliary Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 . . . . § 3 Theorem on the Existence and Uniqueness of Solutions . . 232 . . . . § 4 Smoothness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 . . . . § 5 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . 246 . . . . § 6 Global Attractor and Inertial Sets . . . . . . . . . . . . . . . . . . . . 254 . . . . § 7 Conditions of Regularity of Attractor . . . . . . . . . . . . . . . . . . 261 . . . . § 8 On Singular Limit in the Problem of Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 . . . . § 9 On Inertial and Approximate Inertial Manifolds . . . . . . . . . 276 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Contents 5 Chapter 5 . Theory of Fun Theory of FunTheory of Fun Theory of Funct ctct ctionals ionalsionals ionals th thth that Uniquely Determine Long-Time Dynamics at Uniquely Determine Long-Time Dynamicsat Uniquely Determine Long-Time Dynamics at Uniquely Determine Long-Time Dynamics . . . . § 1 Concept of a Set of Determining Functionals . . . . . . . . . . . 285 . . . . § 2 Completeness Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 . . . . § 3 Estimates of Completeness Defect in Sobolev Spaces . . . . 306 . . . . § 4 Determining Functionals for Abstract Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . 317 . . . . § 5 Determining Functionals for Reaction-Diffusion Systems . . 328 . . . . § 6 Determining Functionals in the Problem of Nerve Impulse Transmission . . . . . . . . . . . . . . . . . . . . . . 339 . . . . § 7 Determining Functionals for Second Order in Time Equations . . . . . . . . . . . . . . . . . . 350 . . . . § 8 On Boundary Determining Functionals . . . . . . . . . . . . . . . . 358 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Chapter 6 . Homoclinic Chaos Homoclinic ChaosHomoclinic Chaos Homoclinic Chaos in Infinite-Dimensional Sy in Infinite-Dimensional Syin Infinite-Dimensional Sy in Infinite-Dimensional Syst stst stems emsems ems . . . . § 1 Bernoulli Shift as a Model of Chaos . . . . . . . . . . . . . . . . . . . 365 . . . . § 2 Exponential Dichotomy and Difference Equations . . . . . . . 369 . . . . § 3 Hyperbolicity of Invariant Sets for Differentiable Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 377 . . . . § 4 Anosov’s Lemma on -trajectories . . . . . . . . . . . . . . . . . . . 381 . . . . § 5 Birkhoff-Smale Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 . . . . § 6 Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . 396 . . . . § 7 On the Existence of Transversal Homoclinic Trajectories . . 402 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 A Палкой щупая дорогу, Бродит наугад слепой, Осторожно ставит ногу И бормочет сам с собой. И на бельмах у слепого Полный мир отображен: Дом, лужок, забор, корова, Клочья неба голубого — Все, чего не видит он. Вл. Ходасевич «Слепой» A blind man tramps at random touching the road with a stick. He places his foot carefully and mumbles to himself. The whole world is displayed in his dead eyes. There are a house, a lawn, a fence, a cow and scraps of the blue sky — everything he cannot see. Vl. Khodasevich « A Blind Man » Preface PrefacePreface Preface The recent years have been marked out by an evergrowing interest in the research of qualitative behaviour of solutions to nonlinear evolutionary partial differential equations. Such equations mostly arise as mathematical models of processes that take place in real (physical, chemical, biological, etc.) systems whose states can be characterized by an infinite number of parameters in general. Dissipative systems form an important class of sys- tems observed in reality. Their main feature is the presence of mechanisms of energy reallocation and dissipation. Interaction of these two mecha- nisms can lead to appearance of complicated limit regimes and structures in the system. Intense interest to the infinite-dimensional dissipative sys- tems was significantly stimulated by attempts to find adequate mathemati- cal models for the explanation of turbulence in liquids based on the notion of strange (irregular) attractor. By now significant progress in the study of dynamics of infinite-dimensional dissipative systems have been made. Moreover, the latest mathematical studies offer a more or less common line (strategy), which when followed can help to answer a number of principal questions about the properties of limit regimes arising in the system under consideration. Although the methods, ideas and concepts from finite-di- mensional dynamical systems constitute the main source of this strategy, finite-dimensional approaches require serious revaluation and adaptation. The book is devoted to a systematic introduction to the scope of main ideas, methods and problems of the mathematical theory of infinite-dimen- sional dissipative dynamical systems. Main attention is paid to the systems that are generated by nonlinear partial differential equations arising in the modern mechanics of continua. The main goal of the book is to help the reader to master the basic strategies of the theory and to qualify him/her for an independent scientific research in the given branch. We also hope that experts in nonlinear dynamics will find the form many fundamental facts are presented in convenient and practical. The core of the book is composed of the courses given by the author at the Department of Mechanics and Mathematics at Kharkov University dur- ing several years. The book consists of 6 chapters. Each chapter corre- sponds to a term course (34-36 hours) approximately. Its body can be inferred from the table of contents. Every chapter includes a separate list of references. The references do not claim to be full. The lists consist of the publications referred to in this book and offer additional works recommen- 8 Preface ded for further reading. There are a lot of exercises in the book. They play a double role. On the one hand, proofs of some statements are presented as (or contain) cycles of exercises. On the other hand, some exercises contain an additional information on the object under consideration. We recom- mend that the exercises should be read at least. Formulae and statements have double indexing in each chapter (the first digit is a section number). When formulae and statements from another chapter are referred to, the number of the corresponding chapter is placed first. It is sufficient to know the basic concepts and facts from functional analysis and ordinary differential equations to read the book. It is quite un- derstandable for under-graduate students in Mathematics and Physics. I.D. Chueshov Chapter 1 Basic Concepts of the Theory Basic Concepts of the TheoryBasic Concepts of the Theory Basic Concepts of the Theory of Infinite-Dimensional of Infinite-Dimensionalof Infinite-Dimensional of Infinite-Dimensional Dynamical Systems Dynamical Systems Dynamical Systems Dynamical Systems Contents . . . . § 1 Notion of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . § 2 Trajectories and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . § 3 Definition of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . § 4 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . . 24 . . . . § 5 Theorems on Existence of Global Attractor . . . . . . . . . . . . . . 28 . . . . § 6 On the Structure of Global Attractor . . . . . . . . . . . . . . . . . . . 34 . . . . § 7 Stability Properties of Attractor and Reduction Principle . . . 45 . . . . § 8 Finite Dimensionality of Invariant Sets . . . . . . . . . . . . . . . . . 52 . . . . § 9 Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems . . . . . . . . . . . . . . . 61 . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 [...]... main result Theorem 5.1 Assume that a dynamical system ( X , St ) is dissipative and asymptotically compact Let B be a bounded absorbing set of the system ( X , St ) Then the set A = w ( B ) is a nonempty compact set and is a global attractor of the dynamical system ( X , St ) The attractor A is a connected set in X In particular, this theorem is applicable to the dynamical systems from Exercises 4.24.11... biology, economics and sociology The notion of dynamical system is the key and uniting element in synergetics Its usage enables us to cover a rather wide spectrum of problems arising in particular sciences and to work out universal approaches to the description of qualitative picture of real phenomena in the universe 12 C h a p t e r 1 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems... the element y = { yi } , where yi = x i + 1 As a result, a dynamical system ( X , S n ) comes into being It is used for describing complicated (quasirandom) behaviour in some quite realistic systems Notion of Dynamical System In the example below we describe one of the approaches that enables us to connect dynamical systems to nonautonomous (and nonperiodic) ordinary differential equations E x a m p... initial state v To emphasize this circumstance, we often write g + = g + ( v ) In general, it is impossible to continue this semitrajectory g + ( v ) to a full trajectory without imposing any additional conditions on the dynamical system Assume that an evolutionary operator St is invertible for some t > 0 Then it is invertible for all t > 0 and for any v ẻ X there exists a negative semitrajectory g =... attractors of infinite-dimensional dissipative dynamical systems a great role is played by the property of asymptotic compactness For the sake of simplicity let us assume that X is a closed subset of a Banach space The dynamical system ( X , St ) is said to be asymptotically compact if for any t > 0 its evolutionary operator St can be expressed by the form (1) St = St where the mappings St( 1 ) and St( 2... present some ideas and approaches of the theory of dynamical systems which are of general-purpose use and applicable to the systems generated by nonlinear partial differential equations Đ1 Notion of Dynamical System In this book dynamical system is taken to mean the pair of objects ( X , St ) consisting of a complete metric space X and a family St of continuous mappings of the space X into itself with... equivalence to the metric d* ( x , Ơ y) = ồ2 i = -Ơ -i xi - yi Trajectories and Invariant Sets Đ2 Trajectories and Invariant Sets Let ( X , St ) be a dynamical system with continuous or discrete time Its trajectory (or orbit ) is defined as a set of the type g = {u (t) : t ẻ T} , where u ( t ) is a continuous function with values in X such that St u ( t ) = u ( t + t ) for all t ẻ T + and t ẻ T Positive... define an operator St in the space X = C [ - 1 , 0 ] by the formula ( St f ) ( t ) = x ( t + t ) , t ẻ [ -1 , 0 ] , where x ( t ) is the solution to problem (1.6) and (1.7), then we obtain an infinite-dimensional dynamical system ( C [ - 1 , 0 ] , St ) 13 14 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems C h a p t e r Now we give several examples of discrete dynamical systems... called the a -limit set for A Trajectories and Invariant Sets Lemma 2.1 For an element y to belong to an w -limit set w ( A) , it is necessary and sufficient that there exist a sequence of elements { yn } è A and a sequence of numbers tn , the latter tending to infinity such that lim d ( St yn , y ) =  , n nđƠ where d ( x , y ) is the distance between the elements x and y in the space X Proof Let the... real number and l is a continuous linear functional on X It is clear that the concepts of global and global weak attractors coincide in the finite-dimensional case In general, a global attractor A is also a global weak attractor, provided the set A is weakly closed E x e r c i s e 3.1 Let A be a global or global weak attractor of a dynamical system ( X , St ) Then it is uniquely determined and contains . Chueshov Introduction IntroductionIntroduction Introduction to the Theory of Infinite-Dimensional to the Theory of Infinite-Dimensionalto the Theory of Infinite-Dimensional to the Theory of Infinite-Dimensional Dissipative. ¥-= ¥ å = Trajectories and Invariant Sets 17 § 2 Trajectories and Invariant Sets § 2 Trajectories and Invariant Sets§ 2 Trajectories and Invariant Sets § 2 Trajectories and Invariant Sets Let be a dynamical. Theory e Theorye Theory e Theory of Infinite-Dimensional Dynamical Sy of Infinite-Dimensional Dynamical Syof Infinite-Dimensional Dynamical Sy of Infinite-Dimensional Dynamical Syst stst stems emsems ems