Kinetic Boltzmann, Vlasov and Related Equations

Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in 1938 and serves as a basis of plasma physics and describes large-scale processes and galaxies in astronomy, star wind theory.

This book provides a comprehensive review of both equations and presents both classical and modern applications. In addition, it discusses several open problems of great importance.

1. Principal Concepts of Kinetic Equations
2. Lagrangian Coordinates
3. Vlasov-Maxwell and Vlasov-Einstein Equations
4. Energetic Substitution
5. Introduction in Mathematical Theory of Kinetic Equations
6. On the Family of the Steady-State Solutions of Vlasov-Maxwell System
7. Boundary Value Problems for the Vlasov-Maxwell System
8. Bifurcation of Stationary Solutions of the Vlasov-Maxwell System
9. Boltzmann Equation
10. Discrete Models of Boltzmann Equation
11. Method of Spherical Harmonics and Relaxation of Maxwellian Gas
12. Discrete Boltzmann equation Models for Mixtures
13. Quantum Hamiltonians and Kinetic Equations
14. Modelling of the Limit Problem for the Magnetically Noninsulated Diode
15. Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods