稀薄气体的数学理论（影印版） 下载 mobi epub pdf 电子书 2024

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内容简介

　　 《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。

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目录

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

精彩书摘

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

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