《稀薄氣體的數學理論》是2009年高等教育出版社出版的圖書，作者是切爾奇納尼。本書講述了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。
基本介紹
- 書名：稀薄氣體的數學理論
- 作者：切爾奇納尼
- ISBN：9787040255355
- 定價：28.40
- 出版社：高等教育出版社
- 出版時間：2009年02月
- 開本：16
內容簡介,圖書目錄,
內容簡介
直到1 994年的理論發展，包括BoItzmann方程是怎樣從經典力學推出來的，即BoItzmann方程是怎樣從Liouville方程推出來的；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方程推出來的。
圖書目錄
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
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
