稀薄氣體的數(shù)學(xué)理論（影印版）

前言

為了更好地借鑒國(guó)外數(shù)學(xué)教育與研究的成功經(jīng)驗(yàn)，促進(jìn)我國(guó)數(shù)學(xué)教育與研究事業(yè)的發(fā)展，提高高等學(xué)校數(shù)學(xué)教育教學(xué)質(zhì)量，本著“為我國(guó)熱愛數(shù)學(xué)的青年創(chuàng)造一個(gè)較好的學(xué)習(xí)數(shù)學(xué)的環(huán)境”這一宗旨，天元基金贊助出版“天元基金影印數(shù)學(xué)叢書”。該叢書主要包含國(guó)外反映近代數(shù)學(xué)發(fā)展的純數(shù)學(xué)與應(yīng)用數(shù)學(xué)方面的優(yōu)秀書籍，天元基金邀請(qǐng)國(guó)內(nèi)各個(gè)方向的知名數(shù)學(xué)家參與選題的工作，經(jīng)專家遴選、推薦，由高等教育出版社影印出版。為了提高我國(guó)數(shù)學(xué)研究生教學(xué)的水平，暫把選書的目標(biāo)確定在研究生教材上。當(dāng)然，有的書也可作為高年級(jí)本科生教材或參考書，有的書則介于研究生教材與專著之間。歡迎各方專家、讀者對(duì)本叢書的選題、印刷、銷售等工作提出批評(píng)和建議。

內(nèi)容概要

本書講述了稀薄氣體的數(shù)學(xué)理論（Boltzmann方程的數(shù)學(xué)理論）中的三個(gè)主要問題直到1994年的理論發(fā)展，包括Boltzmann方程是怎樣從經(jīng)典力學(xué)推出來的，即Boltzmann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(xué)的關(guān)系，即Euler方程和Navier-Stokes方程是怎樣從Liouvi Lle方程推出來的。另外，本書還介紹了O.Lanford III，DiPerna，P.L.Lions等的出色工作，可作為Boltzmann方程的數(shù)學(xué)理論的優(yōu)秀的教材和參考書。

作者簡(jiǎn)介

編者：(意大利)切爾奇納尼(Cereignani,C.)

書籍目錄

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

章節(jié)摘錄

插圖：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 Bernoulli's 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.

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