《稀薄气体的数学理论（影印版）》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BoItzmann方程是怎样从经典力学推出来的，即BoItzmann方程是怎样从Liouville方程推出来的；Boltzmann方程解的存在性问题；Boltzmann方程与流体力学的关系，即EuIer方程和Navier-Stokes方程是怎样从Liouville方程推出来的。另外，《稀薄气体的数学理论（影印版）》还介绍了O．LanfordⅢ，DiPerna，P．L.Lions等的出色工作，可作为BOItzmann方程的数学理论的优秀的教材和参考书。
为了更好地借鉴国外数学教育与研究的成功经验，促进我国数学教育与研究事业的发展，提高高等学校数学教育教学质量，本着“为我国热爱数学的青年创造一个较好的学习数学的环境”这一宗旨，天元基金赞助出版“天元基金影印数学丛书”。
该丛书主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍，天元基金邀请国内各个方向的知名数学家参与选题的工作，经专家遴选、推荐，由高等教育出版社影印出版。为了提高我国数学研究生教学的水平，暂把选书的目标确定在研究生教材上。当然，有的书也可作为高年级本科生教材或参考书，有的书则介于研究生教材与专著之间。
欢迎各方专家、读者对本丛书的选题、印刷、销售等工作提出批评和建议。
Introduction
1 Historical Introduction
1.1 What is a Gas? From the Billiard Table to Boyles Law
1.2 Brief History of Kinetic Theory
2 Informal Derivation of the Boltzmann Equation
2.1 The Phase Space and the Liouville Equation
2.2 Boltzmanns Argument in a Modern Perspective
2.3 Molecular Chaos. Critique and Justification
2.4 The BBGKY Hierarchy
2.5 The Boltzmann Hierarchy and Its Relation to the Boltzmann Equation
3 Elementary Properties of the Solutions
3.1 Collision Invariants 33
3.2 The Boltzmann Inequality and the Maxwell Distributions
3.3 The Macroscopic Balance Equations
3.4 The H-Theorem
3.5 Loschmidts Paradox
3.6 Poincares Recurrence and Zermelos Paradox
3.7 Equilibrium States and Maxwellian Distributions
3.8 Hydrodynamical Limit and Other Scalings
4 Rigorous Validity of the Boltzmann Equation
4.1 Significance of the Problem
4.2 Hard-Sphere Dynamics
4.3 Transition to L1. The Liouville Equation and the BBGKY Hierarchy Revisited
4.4 Rigorous Validity of the Boltzmann Equation
4.5 Validity of the Boltzmann Equation for a Rare Cloud of Gas in the Vacuum
4.6 Interpretation
4.7 The Emergence of Irreversibility
4.8 More 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
5 Existence and Uniqueness Results
5.1 Preliminary Remarks
5.2 Existence from Validity, and Overview
5.3 A General Global Existence Result
5.4 Generalizations and Other Remarks
Appendix 5.A
6 The Initial Value Problem for the Homogeneous Boltzmann Equation
6.1 An Existence Theorem for a Modified Equation
6.2 Removing the Cutoff： The L1-Theory for the Full Equation
6.3 The L∞-Theory and Classical Solutions
6.4 Long Time Behavior
6.5 Further Developments and Comments
Appendix 6.A
Appendix 6.B
Appendix 6.C
7 Perturbations of Equilibria and Space Homogeneous Solutions
7.1 The Linearized Collision Operator
7.2 The Basic Properties of the Linearized Collision Operator
7.3 Spectral Properties of the Fourier-Transformed, Linearized Boltzmann Equation
7.4 The Asymptotic Behavior of the Solution of the Cauchy Problem for the Linearized Boltzmann Equation
7.5 The Global Existence Theorem for the Nonlinear Equation
7.6 Extensions： The Periodic Case and Problems in One and Two Dimensions
7.7 A Further Extension： Solutions Close to a Space Homogeneous Solution
8 Boundary Conditions
8.1 Introduction
8.2 The Scattering Kernel
8.3 The Accommodation Coefficients
8.4 Mathematical Models
8.5 A Remarkable Inequality
9 Existence Results for Initial-Boundary and Boundary Value Problems
9.1 Preliminary Remarks
9.2 Results on the Traces
9.3 Properties of the Free-Streaming Operator
9.4 Existence in a Vessel with Isothermal Boundary
9.5 Rigorous Proof of the Approach to Equilibrium
9.6 Perturbations of Equilibria
9.7 A Steady Problem
9.8 Stability of the Steady Flow Past an Obstacle
9.9 Concluding Remarks
10 Particle Simulation of the Boltzmann Equation
10.1 Rationale amd Overview
10.2 Low Discrepancy Methods
10.3 Birds Scheme
11 Hydrodynamical Limits
11.1 A Formal Discussion
11.2 The Hilbert Expansion
11.3 The Entropy Approach to the Hydrodynamical Limit
11.4 The Hydrodynamical Limit for Short Times
11.5 Other Scalings and the Incompressible Navier-Stokes Equations
12 Open Problems and New Directions
Author Index
Subject Index
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