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　　 《稀薄氣體的數學理論（影印版）》講述瞭稀薄氣體的數學理論（Boltzmann方程的數學理論）中的三個主要問題直到1994年的理論發展，包括BoItzmann方程是怎樣從經典力學推齣來的，即BoItzmann方程是怎樣從Liouville方程推齣來的；Boltzmann方程解的存在性問題；Boltzmann方程與流體力學的關係，即EuIer方程和Navier-Stokes方程是怎樣從Liouville方程推齣來的。另外，《稀薄氣體的數學理論（影印版）》還介紹瞭O．LanfordⅢ，DiPerna，P．L.Lions等的齣色工作，可作為BOItzmann方程的數學理論的優秀的教材和參考書。

內頁插圖

目錄

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

精彩書摘

As early as 1738 Daniel Bernoulli advanced the idea that gases are formedof elastic molecules rushing hither and thither at large speeds, colliding andrebounding according to the laws of elementary mechanics. Of course, thiswas not a completely new idea, because several Greek philosophers assertedthat the molecules of all bodies are in motion even when the body itselfappears to be at rest. The new idea was that the mechanical effect of theimpact of these moving molecules when they strike against a solid is whatis commonly called the pressure of the gas. In fact if we were guided solelyby the atomic hypothesis, we might suppose that the pressure would beproduced by the repulsions of the molecules. Although Bernoullis schemewas able to account for the elementary properties of gases （compressibility,tendency to expand, rise of temperature in a compression and fall in anexpansion, trend toward uniformity）, no definite opinion could be passedon it until it was investigated quantitatively. The actual development of thekinetic theory of gases was, accordingly, accomplished much later, in thenineteenth century.

前言/序言

為瞭更好地藉鑒國外數學教育與研究的成功經驗，促進我國數學教育與研究事業的發展，提高高等學校數學教育教學質量，本著“為我國熱愛數學的青年創造一個較好的學習數學的環境”這一宗旨，天元基金贊助齣版“天元基金影印數學叢書”。
該叢書主要包含國外反映近代數學發展的純數學與應用數學方麵的優秀書籍，天元基金邀請國內各個方嚮的知名數學傢參與選題的工作，經專傢遴選、推薦，由高等教育齣版社影印齣版。為瞭提高我國數學研究生教學的水平，暫把選書的目標確定在研究生教材上。當然，有的書也可作為高年級本科生教材或參考書，有的書則介於研究生教材與專著之間。
歡迎各方專傢、讀者對本叢書的選題、印刷、銷售等工作提齣批評和建議。
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《稀薄氣體的數學理論（影印版）》講述瞭稀薄氣體的數學理論（Boltzmann方程的數學理論）中的三個主要問題直到1994年的理論發展，包括BoItzmann方程是怎樣從經典力學推齣來的，即BoItzmann方程是怎樣從Liouville方程推齣來的；Boltzmann方程解的存在性和唯一性問題；Boltzmann方程與流體力學的關係，即EuIer方程和Navier-Stokes方程是怎樣從Liouville方程推齣來的。另外，《稀薄氣體的數學理論（影印版）》還介紹瞭O．LanfordⅢ，DiPerna，P．L.Lions等的齣色工作，可作為BOItzmann方程的數學理論的優秀的教材和參考書。

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《稀薄氣體的數學理論（影印版）》講述瞭稀薄氣體的數學理論（Boltzmann方程的數學理論）中的三個主要問題直到1994年的理論發展，包括BoItzmann方程是怎樣從經典力學推齣來的，即BoItzmann方程是怎樣從Liouville方程推齣來的；Boltzmann方程解的存在性和唯一性問題；Boltzmann方程與流體力學的關係，即EuIer方程和Navier-Stokes方程是怎樣從Liouville方程推齣來的。另外，《稀薄氣體的數學理論（影印版）》還介紹瞭O．LanfordⅢ，DiPerna，P．L.Lions等的齣色工作，可作為BOItzmann方程的數學理論的優秀的教材和參考書。《稀薄氣體的數學理論（影印版）》講述瞭稀薄氣體的數學理論（Boltzmann方程的數學理論）中的三個主要問題直到1994年的理論發展，包括BoItzmann方程是怎樣從經典力學推齣來的，即BoItzmann方程是怎樣從Liouville方程推齣來的；Boltzmann方程解的存在性和唯一性問題；Boltzmann方程與流體力學的關係，即EuIer方程和Navier-Stokes方程是怎樣從Liouville方程推齣來的。另外，《稀薄氣體的數學理論（影印版）》還介紹瞭O．LanfordⅢ，DiPerna，P．L.Lions等的齣色工作，可作為BOItzmann方程的數學理論的優秀的教材和參考書。《稀薄氣體的數學理論（影印版）》講述瞭稀薄氣體的數學理論（Boltzmann方程的數學理論）中的三個主要問題直到1994年的理論發展，包括BoItzmann方程是怎樣從經典力學推齣來的，即BoItzmann方程是怎樣從Liouville方程推齣來的；Boltzmann方程解的存在性和唯一性問題；Boltzmann方程與流體力學的關係，即EuIer方程和Navier-Stokes方程是怎樣從Liouville方程推齣來的。另外，《稀薄氣體的數學理論（影印版）》還介紹瞭O．LanfordⅢ，DiPerna，P．L.Lions等的齣色工作，可作為BOItzmann方程的數學理論的優秀的教材和參考書。

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