力学和物理学中的无限维动力系统影印版（第2版）

PrefacetotheSecondEdition
PrefacetotheFirstEdition
GENERALINTRODUCTION.
TheUser'sGuide
Introduction
1.MechanismandDescriptionofChaos.TheFinite-DimensionalCase
2.MechanismandDescriptionofChaos.TheInfinite-DimensionalCase
3.TheGlobalAttractor.ReductiontoFiniteDimension
4.RemarksontheComputationalAspect
5.TheUser'sGuide
CHAPTERI
GeneralResultsandConceptsonInvariantSetsandAttractors
Introduction
1.Semigroups,InvariantSets,andAttractors
1.1.SemigroupsofOperators
1.2.FunctionalInvariantSets
1.3.AbsorbingSetsandAttractors
1.4.ARemarkontheStabilityoftheAttractors
2.ExamplesinOrdinaryDifferentialEquations
2.1.ThePendulum
2.2.TheMineaSystem
2.3.TheLorenzModel
3.FractalInterpolationandAttractors
3.1.TheGeneralFramework
3.2.TheInterpolationProcess
3.3.ProofofTheorem3.1
CHAPTERII
ElementsofFunctionalAnalysis
Introduction
1.FunctionSpaces
1.1.DefinitionoftheSpaces.Notations
1.2.PropertiesofSobolevSpaces
1.3.OtherSobolevSpaces
1.4.FurtherPropertiesofSobolevSpaces
2.LinearOperators
2.1.BilinearFormsandLinearOperators
2.2."Concrete"ExamplesofLinearOperators
3.LinearEvolutionEquationsoftheFirstOrderinTime
3.1.Hypotheses
3.2.AResultofExistenceandUniqueness
3.3.RegularityResults
3.4.Time-DependentOperators
4.LinearEvolutionEquationsoftheSecondOrderinTime
4.1.TheEvolutionProblem
4.2.AnotherResult
4.3.Time-DependentOperators
CHAPTERIII
AttractorsoftheDissipativeEvolutionEquationoftheFirstOrder
inTime:Reaction-DiffusionEquations.FluidMechanicsand
PatternFormationEquations
introduction
1.Reaction-DiffusionEquations
1.1.EquationswithaPolynomialNonlinearity
1.2.EquationswithanInvariantRegion
2.Navier-StokesEquations(n=2)
2.1.TheEquationsandTheirMathematicalSetting
2.2.AbsorbingSetsandAttractors
2.3.ProofofTheorem2.1
3.OtherEquationsinFluidMechanics
3.1.AbstractEquation.GeneralResults
3.2.FluidDrivenbyItsBoundary
3.3.Magnetohydrodynamics(MHD)
3.4.GeophysicalFlows(FlowsonaManifold)
3.5.Thermohydraulics
4.SomePatternFormationEquations
4.1.TheKuramoto-SivashinskyEquation
4.2.TheCahn-HilliardEquation
5.SemilinearEquations
5.1.TheEquations.TheSemigroup
5.2.AbsorbingSetsandAttractors
5.3.ProofofTheorem5.2
6.BackwardUniqueness
6.1.AnAbstractResult
6.2.Applications
CHAPTERIV
AttractorsofDissipativeWaveEquations
Introduction
1.LinearEquations:SummaryandAdditionalResults
1.1.TheGeneralFramework
1.2.ExponentialDecay
1.3.BoundedSolutionsontheRealLine
2.TheSine-GordonEquation
2.1.TheEquationandItsMathematicalSetting
2.2.AbsorbingSetsandAttractors
2.3.OtherBoundaryConditions
3.ANonlinearWaveEquationofRelativisticQuantumMechanics
3.1.TheEquationandItsMathematicalSetting
3.2.AbsorbingSetsandAttractors
4.AnAbstractWaveEquation
4.1.TheAbstractEquation.TheGroupofOperators
4.2.AbsorbingSetsandAttractors
4.3.Examples
4.4.ProofofTheorem4.1(Sketch)
5.TheGinzburg-LandauEquation
5.1.TheEquationsandItsMathematicalSetting
5.2.AbsorbingSetsandAttractors
6.WeaklyDissipativeEquations.I.TheNonlinearSchr6dingerEquation
6.1.TheNonlinearSchr6dingerEquation
6.2.ExistenceandUniquenessofSolution.AbsorbingSets
6.3.DecompositionoftheSemigroup
6.4.ComparisonofzandZforLargeTimes
6.5.ApplicationtotheAttractor.TheMainResult
6.6.DeterminingModes
7.WeaklyDissipativeEquationsII.TheKorteweg-DeVriesEquation
7.1.TheEquationanditsMathematicalSetting
7.2.AbsorbingSetsandAttractors
7.3.RegularityoftheAttractor
7.4.ProofoftheResultsinSection7.1
7.5.ProofofProposition7.2
8.UnboundedCase:TheLackofCompactness
8.1.Preliminaries
8.2.TheGlobalAttractor
9.RegularityofAttractors
9.1.APreliminaryResult
9.2.ExampleofPartialRegularity
9.3.ExampleofRegularity
10.StabilityofAttractors
CHAPTERV
LyapunovExponentsandDimensionofAttractors
Introduction
1.LinearandMultilinearAlgebra
1.1.ExteriorProductofHilbertSpaces
1.2.MultilinearOperatorsandExteriorProducts
1.3.ImageofaBallbyaLinearOperator
2.LyapunovExponentsandLyapunovNumbers
2.1.DistortionofVolumesProducedbytheSemigroup
2.2.DefinitionoftheLyapunovExponentsandLyapunovNumbers
2.3.EvolutionoftheVolumeElementandItsExponentialDecay:
TheAbstractFramework
3.HausdorffandFractalDimensionsofAttractors
3.1.HausdorffandFractalDimensions
3.2.CoveringLemmas
3.3.TheMainResults
3.4.ApplicationtoEvolutionEquations
CHAPTERVI
ExplicitBoundsontheNumberofDegreesofFreedomandthe
DimensionofAttractorsofSomePhysicalSystems
Introduction
1.TheLorenzAttractor
2.Reaction-DiffusionEquations
2.1.EquationswithaPolynomialNonlinearity
2.2.EquationswithanInvariantRegion
3.Navier-StokesEquations(n=2)
3.1.GeneralBoundaryConditions
3.2.ImprovementsfortheSpace-PeriodicCase
4.OtherEquationsinFluidMechanics
4.1.TheLinearizedEquations(TheAbstractFramework)
4.2.FluidDrivenbyItsBoundary
4.3.Magnetohydrodynamics
4.4.FlowsonaManifold
4.5.Thermohydraulics
5.PatternFormationEquations
5.1.TheKuramoto-SivashinskyEquation
5.2.TheCahn-HilliardEquations
6.DissipativeWaveEquations
6.1.TheLinearizedEquation
6.2.DimensionoftheAttractor
6.3.Sine-GordonEquations
6.4.SomeLemmas
7.TheGinzburg-LandauEquation
7.1.TheLinearizedEquation
7.2.DimensionoftheAttractor
8.DifferentiabilityoftheSemigroup
CHAPTERVII
Non-Well-PosedProblems,UnstableManifolds,Lyapunov
Functions,andLowerBoundsonDimensions
Introduction
PARTA:NoN-WELL-POSEDPROBLEMS
1.DissipativityandWellPosedness
1.1.GeneralDefinitions
1.2.TheClassofProblemsStudied
1.3.TheMainResult
2.EstimateofDimensionforNon-Well-PosedProblems:
ExamplesinFluidDynamics
2.1.TheEquationsandTheirLinearization
2.2.EstimateoftheDimensionofX
2.3.TheThree-DimensionalNavier-StokesEquations
PARTB:UNSTABLEMANIFOLDS,LYAPUNOVFUNCTIONS,ANDLOWER
BOUNDSONDIMENSIONS
3.StableandUnstableManifolds
3.1.StructureofaMappingintheNeighborhoodofaFixedPoint
3.2.ApplicationtoAttractors
3.3.UnstableManifoldoraCompactInvariantSet
4.TheAttractorofaSemigroupwithaLyapunovFunction
4.1.AGeneralResult
4.2.AdditionalResults
4.3.Examples
5.LowerBoundsonDimensionsofAttractors:AnExample
CHAPTERVIII
TheConeandSqueezingProperties.InertialManifolds
Introduction
1.TheConeProperty
1.1.TheConeProperty
1.2.Generalizations
1.3.TheSqueezingProperty
2.ConstructionofanInertialManifold:DescriptionoftheMethod
2.1.InertialManifolds:TheMethodofConstruction
2.2.TheInitialandPreparedEquations
2.3.TheMapping
3.ExistenceofanInertialManifold
3.1.TheResultofExistence
3.2.FirstPropertiesof
3.3.UtilizationoftheConeProperty
3.4.ProofofTheorem3.1(End)
3.5.AnotherFormofTheorem3.1
4.Examples
4.1.Example1:TheKuramoto-SivashinskyEquation
4.2.Example2:ApproximateInertialManifoldsforthe
Navier-StokesEquations
4.3.Example3:Reaction-DiffusionEquations
4.4.Example4:TheGinzburg-LandauEquation
5.ApproximationandStabilityoftheInertialManifoldwith
RespecttoPerturbations
CHAPTERIX
InertialManifoldsandSlowManifolds.TheNon-Self-AdjointCase
Introduction
1.TheFunctionalSetting
1.1.NotationsandHypotheses
1.2.ConstructionoftheInertialManifold
2.TheMainResult(LipschitzCase)
2.1.ExistenceofInertialManifolds
2.2.Propertiesof
2.3.SmoothnessPropertyof
2.4.ProofofTheorem2.1
3.ComplementsandApplications
3.1.TheLocallyLipschitzCase
3.2.DimensionoftheInertialManifold
4.InertialManifoldsandSlowManifolds
4.1.TheMotivation
4.2.TheAbstractEquation
4.3.AnEquationofNavier-StokesType
CHAPTERX
ApproximationofAttractorsandInertialManifolds.
ConvergentFamiliesofApproximateInertialManifolds
Introduction
1.ConstructionoftheManifolds
1.1.ApproximationoftheDifferentialEquation
1.2.TheApproximateManifolds
2.ApproximationofAttractors
2,1.Propertiesof
2.2.DistancetotheAttractor
2.3.TheMainResult
3.ConvergentFamiliesofApproximateInertialManifolds
3.1.Propertiesof
3.2.DistancetotheExactInertialManifold
3.3.ConvergencetotheExactInertialManifold
APPENDIX
CollectiveSobolevInequalities
Introduction
1.NotationsandHypotheses
1.1.TheOperator
1.2.TheSchrodinger-TypeOperators
2.SpectralEstimatesforSchrodinger-TypeOperators
2.1.TheBirman-SchwingerInequality
2.2.TheSpectralEstimate
3.GeneralizationoftheSobolev-Lieb-ThirringInequality(I)
4.GeneralizationoftheSobolev-Lieb-ThirringInequality(II)
4.1.TheSpace-PeriodicCase
4.2.TheGeneralCase
4.3.ProofofTheorem4.1
5.Examples
Bibliography
Index