The Boltzmann equation and its applications

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.