Discovering Evolution Equations with Applications, Volume 1-Deterministic Equations (Chapman & Hall CRC Applied Mathematics & Nonlinear Science)

458 Pages·2010·2.418 MB·English

by  Mark A. McKibben

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 Paszy´nski, 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 Paszy´nski, 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
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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