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Discovering Evolution Equations with Applications Volume 1-Deterministic Equations C9207_FM.indd 1 6/22/10 1:15:00 PM CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES CHAPMAN & HALL/CRC APPLIED MATHEMATICS Series Editors Goong Chen and Thomas J. Bridges AND NONLINEAR SCIENCE SERIES Published Titles Advanced Differential Quadrature Methods, Zhi Zong and Yingyan Zhang Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 1, One and Two Dimensional Elliptic and Maxwell Problems, Leszek Demkowicz Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 2, Frontiers: Three Discovering Evolution Equations Dimensional Elliptic and Maxwell Problems with Applications, Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszyński, Waldemar Rachowicz, and Adam Zdunek CRC Standard Curves and Surfaces with Mathematica®: Second Edition, with Applications David H. von Seggern Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations, Mark A. McKibben Volume 1-Deterministic Equations Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Victor A. Galaktionov and Sergey R. Svirshchevskii Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Victor A. Galaktionov Introduction to Fuzzy Systems, Guanrong Chen and Trung Tat Pham Introduction to non-Kerr Law Optical Solitons, Anjan Biswas and Swapan Konar Introduction to Partial Differential Equations with MATLAB®, Matthew P. Coleman Introduction to Quantum Control and Dynamics, Domenico D’Alessandro Mathematical Methods in Physics and Engineering with Mathematica, Ferdinand F. Cap Mathematical Theory of Quantum Computation, Goong Chen and Zijian Diao Mathematics of Quantum Computation and Quantum Technology, Goong Chen, Louis Kauffman, and Samuel J. Lomonaco Mixed Boundary Value Problems, Dean G. Duffy Mark A. McKibben Multi-Resolution Methods for Modeling and Control of Dynamical Systems, Puneet Singla and John L. Junkins Goucher College Optimal Estimation of Dynamic Systems, John L. Crassidis and John L. Junkins Quantum Computing Devices: Principles, Designs, and Analysis, Goong Chen, Baltimore, Maryland David A. Church, Berthold-Georg Englert, Carsten Henkel, Bernd Rohwedder, Marlan O. Scully, and M. Suhail Zubairy A Shock-Fitting Primer, Manuel D. Salas Stochastic Partial Differential Equations, Pao-Liu Chow C9207_FM.indd 2 6/22/10 1:15:01 PM CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES CHAPMAN & HALL/CRC APPLIED MATHEMATICS Series Editors Goong Chen and Thomas J. Bridges AND NONLINEAR SCIENCE SERIES Published Titles Advanced Differential Quadrature Methods, Zhi Zong and Yingyan Zhang Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 1, One and Two Dimensional Elliptic and Maxwell Problems, Leszek Demkowicz Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 2, Frontiers: Three Discovering Evolution Equations Dimensional Elliptic and Maxwell Problems with Applications, Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszyński, Waldemar Rachowicz, and Adam Zdunek CRC Standard Curves and Surfaces with Mathematica®: Second Edition, with Applications David H. von Seggern Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations, Mark A. McKibben Volume 1-Deterministic Equations Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Victor A. Galaktionov and Sergey R. Svirshchevskii Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Victor A. Galaktionov Introduction to Fuzzy Systems, Guanrong Chen and Trung Tat Pham Introduction to non-Kerr Law Optical Solitons, Anjan Biswas and Swapan Konar Introduction to Partial Differential Equations with MATLAB®, Matthew P. Coleman Introduction to Quantum Control and Dynamics, Domenico D’Alessandro Mathematical Methods in Physics and Engineering with Mathematica, Ferdinand F. Cap Mathematical Theory of Quantum Computation, Goong Chen and Zijian Diao Mathematics of Quantum Computation and Quantum Technology, Goong Chen, Louis Kauffman, and Samuel J. Lomonaco Mixed Boundary Value Problems, Dean G. Duffy Mark A. McKibben Multi-Resolution Methods for Modeling and Control of Dynamical Systems, Puneet Singla and John L. Junkins Goucher College Optimal Estimation of Dynamic Systems, John L. Crassidis and John L. Junkins Quantum Computing Devices: Principles, Designs, and Analysis, Goong Chen, Baltimore, Maryland David A. Church, Berthold-Georg Englert, Carsten Henkel, Bernd Rohwedder, Marlan O. Scully, and M. Suhail Zubairy A Shock-Fitting Primer, Manuel D. Salas Stochastic Partial Differential Equations, Pao-Liu Chow C9207_FM.indd 3 6/22/10 1:15:01 PM Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-9209-7 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Dedicatedtomywifeandbestfriend,Jodi Contents Preface xiii 1 ABasicAnalysisToolbox 1 1.1 SomeBasicMathematicalShorthand . . . . . . . . . . . . . . . . 1 1.2 SetAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 TheSpace(R,|·|) . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 OrderProperties . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 AbsoluteValue . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 CompletenessPropertyof(R,|·|) . . . . . . . . . . . . . . 7 1.4.4 TopologyofR . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Sequencesin(R,|·|) . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1 SequencesandSubsequences . . . . . . . . . . . . . . . . 12 1.5.2 LimitTheorems. . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.3 CauchySequences . . . . . . . . . . . . . . . . . . . . . . 19 1.5.4 ABriefLookatInfiniteSeries . . . . . . . . . . . . . . . . 21 1.6 TheSpaces(cid:0)RN,(cid:107)·(cid:107) (cid:1)and(cid:16)MN(R),(cid:107)·(cid:107) (cid:17) . . . . . . . . . 24 RN MN(R) 1.6.1 TheSpace(cid:0)RN,(cid:107)·(cid:107) (cid:1) . . . . . . . . . . . . . . . . . . . 25 RN 1.6.1.1 GeometricandTopologicalStructure . . . . . . . 25 1.6.1.2 SequencesinRN . . . . . . . . . . . . . . . . . 28 (cid:16) (cid:17) 1.6.2 TheSpace MN(R),(cid:107)·(cid:107) . . . . . . . . . . . . . . . 29 MN(R) 1.7 AbstractSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7.1 BanachSpaces . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7.2 HilbertSpaces . . . . . . . . . . . . . . . . . . . . . . . . 37 1.8 ElementaryCalculusinAbstractSpaces . . . . . . . . . . . . . . . 41 1.8.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.8.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8.3 TheDerivative . . . . . . . . . . . . . . . . . . . . . . . . 46 1.8.4 “The”Integral . . . . . . . . . . . . . . . . . . . . . . . . 48 1.9 SomeElementaryODEs . . . . . . . . . . . . . . . . . . . . . . . 52 1.9.1 SeparationofVariables . . . . . . . . . . . . . . . . . . . . 52 1.9.2 First-OrderLinearODEs . . . . . . . . . . . . . . . . . . . 53 1.9.3 Higher-OrderLinearODEs. . . . . . . . . . . . . . . . . . 54 1.10 LookingAhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.11 GuidanceforExercises . . . . . . . . . . . . . . . . . . . . . . . . 55 1.11.1 Level1:ANudgeinaRightDirection . . . . . . . . . . . . 55 vii viii DiscoveringEvolutionEquations 1.11.2 Level2:AnAdditionalThrustinaRightDirection . . . . . 61 2 HomogenousLinearEvolutionEquationsinRN 69 2.1 MotivationbyModels . . . . . . . . . . . . . . . . . . . . . . . . 69 2.2 TheMatrixExponential . . . . . . . . . . . . . . . . . . . . . . . 73 2.3 TheHomogenousCauchyProblem:Well-Posedness . . . . . . . . 82 2.4 PerturbationandConvergenceResults . . . . . . . . . . . . . . . . 85 2.5 AGlimpseatLong-TermBehavior . . . . . . . . . . . . . . . . . 87 2.6 LookingAhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.7 GuidanceforExercises . . . . . . . . . . . . . . . . . . . . . . . . 91 2.7.1 Level1:ANudgeinaRightDirection . . . . . . . . . . . . 91 2.7.2 Level2:AnAdditionalThrustinaRightDirection . . . . . 93 3 AbstractHomogenousLinearEvolutionEquations 97 3.1 LinearOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1.1 BoundedversusUnboundedOperators . . . . . . . . . . . 97 3.1.2 InvertibleOperators . . . . . . . . . . . . . . . . . . . . . 102 3.1.3 ClosedOperators . . . . . . . . . . . . . . . . . . . . . . . 103 3.1.4 Densely-Definedoperators . . . . . . . . . . . . . . . . . . 104 3.2 MotivationbyModels . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3 IntroducingSemigroups . . . . . . . . . . . . . . . . . . . . . . . 116 3.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.3.2 UniformlyContinuousSemigroups . . . . . . . . . . . . . 120 3.3.3 StronglyContinuousSemigroups . . . . . . . . . . . . . . 123 3.4 TheAbstractHomogenousCauchyProblem . . . . . . . . . . . . 126 3.5 GenerationTheorems . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.5.1 Hille-YosidaandFeller–Miyadera–PhillipsTheorems. . . . 131 3.5.2 AFirstLookatDissipativeOperators . . . . . . . . . . . . 142 3.6 AUsefulPerturbationResult . . . . . . . . . . . . . . . . . . . . . 145 3.7 SomeApproximationTheory . . . . . . . . . . . . . . . . . . . . 147 3.8 ABriefGlimpseatLong-TermBehavior . . . . . . . . . . . . . . 149 3.9 AnImportantLookBack . . . . . . . . . . . . . . . . . . . . . . . 150 3.10 LookingAhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.11 GuidanceforExercises . . . . . . . . . . . . . . . . . . . . . . . . 153 3.11.1 Level1:ANudgeinaRightDirection . . . . . . . . . . . . 153 3.11.2 Level2:AnAdditionalThrustinaRightDirection . . . . . 157 4 NonhomogenousLinearEvolutionEquations 163 4.1 Finite-DimensionalSetting . . . . . . . . . . . . . . . . . . . . . . 163 4.1.1 MotivationbyModels . . . . . . . . . . . . . . . . . . . . 163 4.1.2 One-DimensionalCase . . . . . . . . . . . . . . . . . . . . 165 4.1.3 ExtensionofTheorytoRN . . . . . . . . . . . . . . . . . . 169 4.2 Infinite-DimensionalSetting . . . . . . . . . . . . . . . . . . . . . 171 4.2.1 MotivationbyModels . . . . . . . . . . . . . . . . . . . . 171 4.2.2 TheoryinaGeneralBanachSpaceX . . . . . . . . . . . . 172 Contents ix 4.3 IntroducingTwoNewModels . . . . . . . . . . . . . . . . . . . . 177 4.4 LookingAhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.5 GuidanceforExercises . . . . . . . . . . . . . . . . . . . . . . . . 185 4.5.1 Level1:ANudgeinaRightDirection . . . . . . . . . . . . 185 4.5.2 Level2:AnAdditionalThrustinaRightDirection . . . . . 188 5 Semi-LinearEvolutionEquations 191 5.1 MotivationbyModels . . . . . . . . . . . . . . . . . . . . . . . . 191 5.1.1 SomeModelsRevisited . . . . . . . . . . . . . . . . . . . 191 5.1.2 IntroducingTwoNewModels . . . . . . . . . . . . . . . . 192 5.2 MoreToolsfromFunctionalAnalysis . . . . . . . . . . . . . . . . 195 5.2.1 Fixed-PointTheory . . . . . . . . . . . . . . . . . . . . . . 195 5.2.1.1 TheContractionMappingPrinciple . . . . . . . . 195 5.2.1.2 Schauder’sFixedPointTheorem . . . . . . . . . 197 5.2.1.3 CompactOperatorsandSchaefer’sFixed-PointThe- orem . . . . . . . . . . . . . . . . . . . . . . . . 199 5.2.1.4 TheFixed-PointApproach . . . . . . . . . . . . 202 5.2.2 AHandfulofIntegralInequalities . . . . . . . . . . . . . . 202 5.2.3 FrechetDifferentiability . . . . . . . . . . . . . . . . . . . 205 5.3 SomeEssentialPreliminaryConsiderations . . . . . . . . . . . . . 206 5.4 GrowthConditions . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.5 TheoryforLipschitz-TypeForcingTerms . . . . . . . . . . . . . . 212 5.5.1 ExistenceandUniquenessResults . . . . . . . . . . . . . . 212 5.5.2 ContinuousDependence . . . . . . . . . . . . . . . . . . . 224 5.5.3 ExtendabilityofLocalSolutions . . . . . . . . . . . . . . . 226 5.5.4 Long-TermBehavior . . . . . . . . . . . . . . . . . . . . . 229 5.5.5 ModelsRevisited . . . . . . . . . . . . . . . . . . . . . . . 230 5.6 TheoryforNon-Lipschitz-TypeForcingTerms . . . . . . . . . . . 236 5.7 TheoryunderCompactnessAssumptions . . . . . . . . . . . . . . 243 5.8 ASummarizingLookBack . . . . . . . . . . . . . . . . . . . . . 250 5.9 LookingAhead . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.10 GuidanceforExercises . . . . . . . . . . . . . . . . . . . . . . . . 252 5.10.1 Level1:ANudgeinaRightDirection . . . . . . . . . . . . 252 5.10.2 Level2:AnAdditionalThrustinaRightDirection . . . . . 257 6 FunctionalEvolutionEquations 263 6.1 MotivationbyModels . . . . . . . . . . . . . . . . . . . . . . . . 263 6.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 6.3 TheoryintheLipschitzCase . . . . . . . . . . . . . . . . . . . . . 272 6.4 TheoryunderCompactnessAssumptions . . . . . . . . . . . . . . 275 6.5 Models–NewandOld . . . . . . . . . . . . . . . . . . . . . . . . 276 6.6 LookingAhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.7 GuidanceforExercises . . . . . . . . . . . . . . . . . . . . . . . . 290 6.7.1 Level1:ANudgeinaRightDirection . . . . . . . . . . . . 290 6.7.2 Level2:AnAdditionalThrustinaRightDirection . . . . . 295

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