稀薄气体的数学理论

Introduction

1HistoricalIntroduction

1.1WhatisaGas?FromtheBilliardTabletoBoylesLaw

1.2BriefHistoryofKineticTheory

2InformalDerivationoftheBoltzmannEquation

2.1ThePhaseSpaceandtheLiouvilleEquation

2.2BoltzmannsArgumentinaModernPerspective

2.3MolecularChaos.CritiqueandJustification

2.4TheBBGKYHierarchy

2.5TheBoltzmannHierarchyandItsRelationtotheBoltzmannEquation

3ElementaryPropertiesoftheSolutions

3.1CollisionInvariants33

3.2TheBoltzmannInequalityandtheMaxwellDistributions

3.3TheMacroscopicBalanceEquations

3.4TheH-Theorem

3.5LoschmidtsParadox

3.6PoincaresRecurrenceandZermelosParadox

3.7EquilibriumStatesandMaxwellianDistributions

3.8HydrodynamicalLimitandOtherScalings

4RigorousValidityoftheBoltzmannEquation

4.1SignificanceoftheProblem

4.2Hard-SphereDynamics

4.3TransitiontoL1.TheLiouvilleEquationandtheBBGKYHierarchyRevisited

4.4RigorousValidityoftheBoltzmannEquation

4.5ValidityoftheBoltzmannEquationforaRareCloudofGasintheVacuum

4.6Interpretation

4.7TheEmergenceofIrreversibility

4.8MoreontheBoltzmannHierarchy

Appendix4.AMoreaboutHard-SphereDynamics

Appendix4.BARigorousDerivationoftheBBGKYHierarchy

Appendix4.CUchiyamasExample

5ExistenceandUniquenessResults

5.1PreliminaryRemarks

5.2ExistencefromValidity,andOverview

5.3AGeneralGlobalExistenceResult

5.4GeneralizationsandOtherRemarks

Appendix5.A

6TheInitialValueProblemfortheHomogeneousBoltzmannEquation

6.1AnExistenceTheoremforaModifiedEquation

6.2RemovingtheCutoff:TheL1-TheoryfortheFullEquation

6.3TheL∞-TheoryandClassicalSolutions

6.4LongTimeBehavior

6.5FurtherDevelopmentsandComments

Appendix6.A

Appendix6.B

Appendix6.C

7PerturbationsofEquilibriaandSpaceHomogeneousSolutions

7.1TheLinearizedCollisionOperator

7.2TheBasicPropertiesoftheLinearizedCollisionOperator

7.3SpectralPropertiesoftheFourier-Transformed,LinearizedBoltzmannEquation

7.4TheAsymptoticBehavioroftheSolutionoftheCauchyProblemfortheLinearizedBoltzmannEquation

7.5TheGlobalExistenceTheoremfortheNonlinearEquation

7.6Extensions:ThePeriodicCaseandProblemsinOneandTwoDimensions

7.7AFurtherExtension:SolutionsClosetoaSpaceHomogeneousSolution

8BoundaryConditions

8.1Introduction

8.2TheScatteringKernel

8.3TheAccommodationCoefficients

8.4MathematicalModels

8.5ARemarkableInequality

9ExistenceResultsforInitial-BoundaryandBoundaryValueProblems

9.1PreliminaryRemarks

9.2ResultsontheTraces

9.3PropertiesoftheFree-StreamingOperator

9.4ExistenceinaVesselwithIsothermalBoundary

9.5RigorousProofoftheApproachtoEquilibrium

9.6PerturbationsofEquilibria

9.7ASteadyProblem

9.8StabilityoftheSteadyFlowPastanObstacle

9.9ConcludingRemarks

10ParticleSimulationoftheBoltzmannEquation

10.1RationaleamdOverview

10.2LowDiscrepancyMethods

10.3BirdsScheme

11HydrodynamicalLimits

11.1AFormalDiscussion

11.2TheHilbertExpansion

11.3TheEntropyApproachtotheHydrodynamicalLimit

11.4TheHydrodynamicalLimitforShortTimes

11.5OtherScalingsandtheIncompressibleNavier-StokesEquations

12OpenProblemsandNewDirections

AuthorIndex

SubjectIndex

　　本书讲述了稀薄气体的数学理论(Boltzmann方程的数学理论)中的三个主要问题直到1994年的理论发展，包括Boltzmann方程是怎样从经典力学推出来的；Boltzmann方程解的存在性和唯一性问题；Boltzmann方程与流体力学的关系。另外，本书还介绍了O.LanfordIII，DiPerna，P.L.Lions等的出色工作。本书内容丰富，讲解通俗易懂，具有很强的可读性。本书为英文影印版。　　本书讲述了稀薄气体的数学理论(Boltzmann方程的数学理论)中的三个主要问题直到1994年的理论发展，包括Boltzmann方程是怎样从经典力学推出来的，即Boltzmann方程是怎样从Liouville方程推出来的；Boltzmann方程解的存在性和唯一性问题；Boltzmann方程与流体力学的关系，即Euler方程和Navier-Stokes方程是怎样从LiouviLle方程推出来的。另外，本书还介绍了O.LanfordIII，DiPerna，P.L.Lions等的出色工作，可作为Boltzmann方程的数学理论的优秀的教材和参考书。