The Boltzmann equation and its applications
The Boltzmann equation and its applications
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DOI：
10.2307/3618229
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发表时间：
1988
期刊：
影响因子：
--
通讯作者：
C. Cercignani
中科院分区：
文献类型：
--
作者：
C. Cercignani
1、气体运动理论的基本原理。- 1。介绍。- 2。概率。- 3。相空间和刘维尔定理。- 4。坚硬的球体和坚硬的墙壁。平均自由程。- 5所示。体元在相空间中的散射。- 6所示。时间平均，遍历假设和平衡状态。——引用。- II。玻尔兹曼方程。- 1。非平衡态的问题。- 2。刚性球体气体的多粒子分布函数方程。- 3。刚性球体的玻尔兹曼方程。- 4。概括。- 5所示。碰撞项的细节。- 6所示。碰撞算子的基本性质。碰撞不变量。- 7所示。方程Q(f,f) = 0的解。- 8。气体动力学的微观描述与宏观描述之间的联系。- 9。非截止势和掠碰。福克尔普朗克方程。- 10。模型方程。——引用。- III。气-表面相互作用和h定理。- 1。边界条件和气面相互作用。- 2。散射核的计算。- 3。互惠。- 4。这是一个显著的不平等。- 5所示。麦克斯韦边界条件。住宿系数。- 6所示。气体表面相互作用的数学模型。- 7所示。气-表面相互作用的物理模型。- 8。分子束的散射。- 9。h定理。不可逆性。- 10。平衡态和麦克斯韦分布。——引用。-四、线性运输。- 1。线性化的碰撞算子。- 2。线性化玻尔兹曼方程。- 3。线性玻尔兹曼方程。中子输运和辐射传递。- 4。初值与边值问题解的唯一性。- 5所示。线性化碰撞项的进一步研究。- 6所示。平衡的衰减和碰撞算子的谱。- 7所示。稳定的一维问题。传输系数。- 8。一般情况下。- 9。线性化动力学模型。- 10。变分原理。- 11。格林函数。- 12。积分方程法。——引用。- V.大小平均自由路径。- 1。克努森数。- 2。希尔伯特膨胀。- 3。查普曼-恩斯科格扩张。- 4。对Chapman-Enskog方法的批评。- 5所示。初始层、边界层和激波层。- 6所示。关于Chapman-Enskog方法和输运系数计算的进一步说明。- 7所示。自由分子流过凸体。- 8。存在非凸边界的自由分子流动。- 9。几乎是自由分子流动。——引用。-六、模型的解析解。- 1。初等解法。- 2。一维模型方程的分裂。- 3。最简单输运方程的初等解。- 4。一般方法在Kramers和Milne问题中的应用。- 5所示。应用于平行板间流动及板坯的临界问题。- 6所示。等碰撞频率动力学模型的非定常解。- 7所示。具体问题的分析解决方案。- 8。更一般的模型。- 9。一些特殊情况。- 10。速度相关碰撞频率动力学模型的非定常解。- 11。分析延续。- 12。单原子气体中的声音传播。- 13。二维和三维问题。流过固体。- 14。波动和光散射。——引用。——七世。过渡制度。- 1。介绍。- 2。矩和离散坐标方法。- 3。变分法。- 4。蒙特卡罗方法。- 5所示。以平面或圆柱体为界的区域内的流动和传热问题。- 6所示。冲击波的结构。- 7所示。外部流。- 8。气体在真空中膨胀。——引用。——八世。玻尔兹曼方程解的定理。- 1。介绍。- 2。空间齐次的情况。- 3。玻尔兹曼方程的缓和版和其他修正版。- 4。玻尔兹曼方程的非标准分析方法。- 5所示。玻尔兹曼方程的局部存在性和有效性。- 6所示。接近平衡的全局存在。- 7所示。真空的微扰。- 8。Homoenergetic解决方案。- 9。边值问题。线性化和弱非线性的情况。- 10。非线性边值问题。- 11。结束语。——引用。——引用。-作者索引。
I. Basic Principles of The Kinetic Theory of Gases.- 1. Introduction.- 2. Probability.- 3. Phase space and Liouville's theorem.- 4. Hard spheres and rigid walls. Mean free path.- 5. Scattering of a volume element in phase space.- 6. Time averages, ergodic hypothesis and equilibrium states.- References.- II. The Boltzmann Equation.- 1. The problem of nonequilibrium states.- 2. Equations for the many particle distribution functions for a gas of rigid spheres.- 3. The Boltzmann equation for rigid spheres.- 4. Generalizations.- 5. Details of the collision term.- 6. Elementary properties of the collision operator. Collision invariants.- 7. Solution of the equation Q(f,f) = 0.- 8. Connection between the microscopic description and the macroscopic description of gas dynamics.- 9. Non-cutoff potentials and grazing collisions. Fokker-Planck equation.- 10. Model equations.- References.- III. Gas-Surface Interaction and the H-Theorem.- 1. Boundary conditions and the gas-surface interaction.- 2. Computation of scattering kernels.- 3. Reciprocity.- 4. A remarkable inequality.- 5. Maxwell's boundary conditions. Accommodation coefficients.- 6. Mathematical models for gas-surface interaction.- 7. Physical models for gas-surface interaction.- 8. Scattering of molecular beams.- 9. The H-theorem. Irreversibility.- 10. Equilibrium states and Maxwellian distributions.- References.- IV, Linear Transport.- 1. The linearized collision operator.- 2. The linearized Boltzmann equation.- 3. The linear Boltzmann equation. Neutron transport and radiative transfer.- 4. Uniqueness of the solution for initial and boundary value problems.- 5. Further investigation of the linearized collision term.- 6. The decay to equilibrium and the spectrum of the collision operator.- 7. Steady one-dimensional problems. Transport coefficients.- 8. The general case.- 9. Linearized kinetic models.- 10. The variational principle.- 11. Green's function.- 12. The integral equation approach.- References.- V. Small and Large Mean Free Paths.- 1. The Knudsen number.- 2. The Hilbert expansion.- 3. The Chapman-Enskog expansion.- 4. Criticism of the Chapman-Enskog method.- 5. Initial, boundary and shock layers.- 6. Further remarks on the Chapman-Enskog method and the computation of transport coefficients.- 7. Free molecule flow past a convex body.- 8. Free molecule flow in presence of nonconvex boundaries.- 9. Nearly free-molecule flows.- References.- VI. Analytical Solutions of Models.- 1. The method of elementary solutions.- 2. Splitting of a one-dimensional model equation.- 3. Elementary solutions of the simplest transport equation.- 4. Application of the general method to the Kramers and Milne problems.- 5. Application to the flow between parallel plates and the critical problem of a slab.- 6. Unsteady solutions of kinetic models with constant collision frequency.- 7. Analytical solutions of specific problems.- 8. More general models.- 9. Some special cases.- 10. Unsteady solutions of kinetic models with velocity dependent collision frequency.- 11. Analytic continuation.- 12. Sound propagation in monatomic gases.- 13. Two-dimensional and three-dimensional problems. Flow past solid bodies.- 14. Fluctuations and light scattering.- References.- VII. The Transition Regime.- 1. Introduction.- 2. Moment and discrete ordinate methods.- 3. The variational method.- 4. Monte Carlo methods.- 5. Problems of flow and heat transfer in regions bounded by planes or cylinders.- 6. Shock-wave structure.- 7. External flows.- 8. Expansion of a gas into a vacuum.- References.- VIII. Theorems on the Solutions of the Boltzmann Equation.- 1. Introduction.- 2. The space homogeneous case.- 3. Mollified and other modified versions of the Boltzmann equation.- 4. Nonstandard analysis approach to the Boltzmann equation.- 5. Local existence and validity of the Boltzmann equation.- 6. Global existence near equilibrium.- 7. Perturbations of vacuum.- 8. Homoenergetic solutions.- 9. Boundary value problems. The linearized and weakly nonlinear cases.- 10. Nonlinear boundary value problems.- 11. Concluding remarks.- References.- References.- Author Index.