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.Lanford III，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Liouvi Lle方程推出来的。另外，本书还介绍了O.Lanford III，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。

Introduction

1 Historical Introduction 

 1.1 What is a Gas? From the Billiard Table to Boyle's 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 Boltzmann's 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 Loschmidt's Paradox 

 3.6 Poincare's Recurrence and Zermelo's 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 Uchiyama's 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 Bird's 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