The Cauchy Problem in Kinetic Theory

This clearly written, self-contained volume studies the basic equations of kinetic theory in all of space. It contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations, including the Boltzmann equation (from rarefied gas dynamics) and the Vlasov-Poisson/Vlasov-Maxwell systems (from plasma physics). This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although these equations describe very different phenomena, they share the same streaming term. The author proves that solutions starting from a given configuration at an initial time exist for all future times by imposing appropriate hypotheses on the initial values in several important cases. He emphasizes those questions that a mathematician would ask first: Is there a solution to this problem? Is it unique? Can it be numerically approximated?

Preface; 1. Properties of the Collision Operator. Kinetic Theory, Derivation of the Equations, The Form of the Collision Operator, The Hard Sphere Case, Conservation Laws and the Entropy, Relevance of the Maxwellian, The Jacobian Determinant, The Structure of Collision Invariants, Relationship of the Boltzmann Equation to the Equations of Fluids, References; 2. The Boltzmann Equation Near the Vacuum. Invariance of $|x-tv|^2+|x-tu|^2$, Sequences of Approximate Solutions, Satisfaction of the Beginning Condition, Proof that $u=\ell$, Remarks and Related Questions, References; 3. The Boltzmann Equation Near the Equilibrium. The Perturbation from Equilibrium, Computation of the Integral Operator, Estimates on the Integral Operator, Properties of $L$, Compactness of $K$, Solution Spaces, An Orthonormal Basis for $N(L)$, Estimates on the Nonlinear Term, Equations for 13 Moments, Computation of the Coefficient Matrices, Compensating Functions, Time Decay Estimates, Time Decay in Other Norms, The Major Theorem, The Relativistic Boltzmann Equation, References; 4. The Vlasov--Poisson System. Introduction, Preliminaries and A Priori Estimates, Sketch of the Existence Proof, The Good, the Bad and the Ugly, The Bound on the Velocity Support, Blow-up in the Gravitational Case, References; 5. The Vlasov--Maxwell System. Collisionless Plasmas, Control of Large Velocities, Representation of the Fields, Representation of the Derivatives of the Fields, Estimates on the Particle Density, Bounds on the Field, Bounds on the Gradient of the Field, Proof of Existence, References; 6. Dilute Collisionless Plasmas. The Small--data Theorem, Outline of the Proof, Characteristics, The Particle Densities, Estimates on the Fields, Estimates on Derivatives of the Fields, References; 7. Velocity Averages: Weak Solutions to the Vlasov--Maxwell System. Sketch of the Problem, The Velocity Averaging Smoothing Effect, Convergence of the Current Density, Completion of the Proof, References; 8. Convergence of a Particle Method for the Vlasov--Maxwell System. Introduction, The Particle Simulation, The Field Errors, The Particle Errors, Summing the Errors, References; Index.

Studies the basic equations of kinetic theory in all of space, and contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations. This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although describing very different phenomena, these equations share the same streaming term.