PrefaceAbout the Authors1. Principal Concepts of Kinetic Equations 1.1. Introduction1.2. Kinetic Equations of Boltzmann Kind1.3. Vlasov's Type Equations1.4. How did the Concept of Distribution Function Explain Molecular-Kinetic and Gas Laws to Maxwell1.5. On a Kinetic Approach to the Sixth Hilbert Problem (Axiomatization of Physics)1.6. Conclusions2. Lagrangian Coordinates 2.1. The Problem of N-Bodies, Continuum of Bodies, and Lagrangian Coordinates in Vlasov Equation2.2. When the Equations for Continuum of Bodies Become Hamiltonian?2.3. Oscillatory Potential Example2.4. Antioscillatory Potential Example2.5. Hydrodynamical Substitution: Multiflow Hydrodynamics and Euler-Lagrange Description2.6. Expanding Universe Paradigm2.7. Conclusions3. Vlasov-Maxwell and Vlasov-Einstein Equations 3.1. Introduction3.2. A Shift of Density Along the Trajectories of Dynamical System3.3. Geodesic Equations and Evolution of Distribution Function on Riemannian Manifold3.4. How does the Riemannian Space Measure Behave While Being Transformed?3.5. Derivation of the Vlasov-Maxwell Equation3.6. Derivation Scheme of Vlasov-Einstein Equation3.7. Conclusion4. Energetic Substitution 4.1. System of Vlasov-Poisson Equations for Plasma and Electrons4.2. Energetic Substitution and Bernoulli Integral4.3. Boundary-Value Problem for Nonlinear Elliptic Equation4.4. Verifying the Condition Ψ′ ≥ 04.5. Conclusions5. Introduction to the Mathematical Theory of Kinetic Equations 5.1. Characteristics of the System5.2. Vlasov-Maxwell and Vlasov-Poisson Systems5.3. Weak Solutions of Vlasov-Poisson and Vlasov-Maxwell Systems5.4. Classical Solutions of VP and VM Systems5.5. Kinetic Equations Modeling Semiconductors5.6. Open Problems for Vlasov-Poisson and Vlasov-Maxwell Systems6. On the Family of the Steady-State Solutions of Vlasov-Maxwell System 6.1. Ansatz of the Distribution Function and Reduction of Stationary Vlasov-Maxwell Equations to Elliptic System6.2. Boundary Value Problem6.3. Solutions with Norm7. Boundary Value Problems for the Vlasov-Maxwell System 7.1. Introduction7.2. Existence and Properties of the Solutions of the Vlasov-Maxwell and Vlasov-Poisson Systems in the Bounded Domains7.3. Existence and Properties of Solutions of the VM System in the Bounded Domains7.4. Collisionless Kinetic Models (Classical and Relativistic Vlasov-Maxwell Systems)7.5. Stationary Solutions of Vlasov-Maxwell System7.6. Existence of Solutions for the Boundary Value Problem (7.5.28)–(7.5.30)7.7. Existence of Solution for Nonlocal Boundary Value Problem7.8. Nonstationary Solutions of the Vlasov-Maxwell System7.9. Linear Stability of the Stationary Solutions of the Vlasov-Maxwell System7.10. Application Examples with Exact Solutions7.11. Normalized Solutions for a One-Component Distribution Function8. Bifurcation of Stationary Solutions of the Vlasov-Maxwell System 8.1. Introduction8.2. Bifurcation of Solutions of Nonlinear Equations in Banach Spaces8.3. Conclusions8.4. Statement of Boundary Value Problem and the Problem on Point of Bifurcation of System (8.4.7), (8.4.13)8.5. Resolving Branching Equation8.6. The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions9. Boltzmann Equation 9.1. Collision Integral9.2. Conservation Laws and H-Theorem9.3. Boltzmann Equation for Mixtures9.4. Quantum Kinetic Equations (Uehling-Uhlenbeck Equations)9.5. Peculiarity of Hydrodynamic Equations, Obtained from Kinetic Equations9.6. Linear Boltzmann Equation and Markovian Processes9.7. Time Averages and Boltzmann Extremals10. Discrete Models of Boltzmann Equation 10.1. General Discrete Models of Boltzmann Equation10.2. Calerman, Godunov-Sultangazin, and Broadwell Models10.3. H- Theorem and Conservation Laws10.4. The Class of Decreasing Functionals for Discrete Models: Uniqueness Theorem of the Boltzmann H- Function10.5. Relaxation Problem10.6. Chemical Kinetics Equations and H- Theorem: Conditions of Chemical Equilibrium11. Method of Spherical Harmonics and Relaxation of Maxwellian Gas 11.1. Linear Operators Commuting with Rotation Group11.2. Bilinear Operators Commuting with Rotation Group11.3. Momentum System and Maxwellian Gas Relaxation to Equilibrium. Bobylev Symmetry11.4. Exponential Series and Superposition of Travelling Waves12. Discrete Boltzmann Equation Models for Mixtures 12.1. Discrete Models with Impulses on the Lattice12.2. Invariants12.3. Inductive Process12.4. On Solution of Diophantine Equations of Conservation Laws and Classification of Collisions12.5. Boltzmann Equation for the Mixture in One-Dimensional Case12.6. Models in One-Dimensional Case12.7. The Models in Two-Dimensional Cases12.8. Conclusions12.9. Photo-, Electro-, Magneto-, and Thermophoresis and Reactive Forces13. Quantum Hamiltonians and Kinetic Equations 13.1. Conservation Laws for Polynomial Hamiltonians13.2. Conservation Laws for Kinetic Equations13.3. The Asymptotics of Spectrum for Hamiltonians of Raman Scattering13.4. The Systems of Special Polynomials in the Problems of Quantum Optics13.5. Representation of General Commutation Relations13.6. Tower of Mathematical Physics13.7. Conclusions14. Modeling of the Limit Problem for the Magnetically Noninsulated Diode 14.1. Introduction14.2. Description of Vacuum Diode14.3. Description of the Mathematical Model14.4. Solution Trajectory, Upper and Lower Solutions14.5. Existence of Solutions for System (14.3.18)–(14.3.22)14.6. Analysis of the Known Upper and Lower Solutions14.7. Conclusions15. Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods 15.1. Introduction15.2. Problem Statement15.3. The Overview of Preceeding Results15.4. Eigen Expansion of Generalized Liouville Operator15.5. Hermitian Function Expansion15.6. Another Application Example for Hermite Polynomial DecompositionGlossary of Terms and SymbolsBibliography