力学和物理学中的无限维动力系统影印版（第2版）

　　Since publication of the first edition of this book in 1988， the study of dynamical systems of infinite dimension has been a very active area in pure and applied mathematics; new results include the study of the existence of attractors for a large number of systems in mathematical physics and mechanics; lower and upper estimates on the dimension of the attractors; approximation of attractors; inertial manifolds and their approximation. The study of multilevel numerical methods stemming from dynamical systems theory has also developed as a subject on its own. Finally， intermediate concepts between attractors and inertial manifolds have also been introduced， in particular the concept of inertial sets.本书为英文版。

暂缺《力学和物理学中的无限维动力系统影印版（第2版）》作者简介

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