內容簡介
稀薄氣體的數學理論圖書目錄
Introduction
1Historical Introduction
1.1What is a Gas? From the Billiard Table to Boyles Law
1.2Brief History of Kinetic Theory
2Informal Derivation of the Boltzmann Equation
2.1The Phase Space and the Liouville Equation
2.2Boltzmanns Argument in a Modern Perspective
2.3Molecular Chaos. Critique and Justification
2.4The BBGKY Hierarchy
2.5The Boltzmann Hierarchy and Its Relation to the Boltzmann Equation
3Elementary Properties of the Solutions
3.1Collision Invariants33
3.2The Boltzmann Inequality and the Maxwell Distributions
3.3The Macroscopic Balance Equations
3.4The H-Theorem
3.5Loschmidts Paradox
3.6Poincares Recurrence and Zermelos Paradox
3.7Equilibrium States and Maxwellian Distributions
3.8Hydrodynamical Limit and Other Scalings
4Rigorous Validity of the Boltzmann Equation
4.1Significance of the Problem
4.2Hard-Sphere Dynamics
4.3Transition to L1. The Liouville Equation and the BBGKY Hierarchy Revisited
4.4Rigorous Validity of the Boltzmann Equation
4.5Validity of the Boltzmann Equation for a Rare Cloud of Gas in the Vacuum
4.6Interpretation
4.7The Emergence of Irreversibility
4.8More on the Boltzmann Hierarchy
Appendix 4.A More about Hard-Sphere Dynamics
Appendix 4.B A Rigorous Derivation of the BBGKY Hierarchy
Appendix 4.C Uchiyamas Example
5Existence and Uniqueness Results
5.1Preliminary Remarks
5.2Existence from Validity, and Overview
5.3A General Global Existence Result
5.4Generalizations and Other Remarks
Appendix 5.A
6The Initial Value Problem for the Homogeneous Boltzmann Equation
6.1An Existence Theorem for a Modified Equation
6.2Removing the Cutoff： The L1-Theory for the Full Equation
6.3The L∞-Theory and Classical Solutions
6.4Long Time Behavior
6.5Further Developments and Comments
Appendix 6.A
Appendix 6.B
Appendix 6.C
7Perturbations of Equilibria and Space Homogeneous Solutions
7.1The Linearized Collision Operator
7.2The Basic Properties of the Linearized Collision Operator
7.3Spectral Properties of the Fourier-Transformed, Linearized Boltzmann Equation
7.4The Asymptotic Behavior of the Solution of the Cauchy Problem for the Linearized Boltzmann Equation
7.5The Global Existence Theorem for the Nonlinear Equation
7.6Extensions： The Periodic Case and Problems in One and Two Dimensions
7.7A Further Extension： Solutions Close to a Space Homogeneous Solution
8Boundary Conditions
8.1Introduction
8.2The Scattering Kernel
8.3The Accommodation Coefficients
8.4Mathematical Models
8.5A Remarkable Inequality
9Existence Results for Initial-Boundary and Boundary Value Problems
9.1Preliminary Remarks
9.2Results on the Traces
9.3Properties of the Free-Streaming Operator
9.4Existence in a Vessel with Isothermal Boundary
9.5Rigorous Proof of the Approach to Equilibrium
9.6Perturbations of Equilibria
9.7A Steady Problem
9.8Stability of the Steady Flow Past an Obstacle
9.9Concluding Remarks
10Particle Simulation of the Boltzmann Equation
10.1Rationale amd Overview
10.2Low Discrepancy Methods
10.3Birds Scheme
11Hydrodynamical Limits
11.1A Formal Discussion
11.2The Hilbert Expansion
11.3The Entropy Approach to the Hydrodynamical Limit
11.4The Hydrodynamical Limit for Short Times
11.5Other Scalings and the Incompressible Navier-Stokes Equations
12Open Problems and New Directions
Author Index
Subject Index
