Robert T. Glassey
"The book will make a very valuable addition to the bookshelf of every researcher in mathematical kinetic theory."  Reinhard Ilner, SIAM Review, Vol. 39, Number 3, September 1997
This clearly written, selfcontained volume studies the basic equations of kinetic theory in all of space. It contains uptodate, stateoftheart treatments of initialvalue problems for the major kinetic equations, including the Boltzmann equation (from rarefied gas dynamics) and the VlasovPoisson/VlasovMaxwell systems (from plasma physics). This is the only existing book to treat Boltzmanntype problems and Vlasovtype 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?
The topics treated include the study of the Boltzmann collision operator, the study of the initialvalue problem for the Boltzmann equation with "small" and "near equilibrium" data, global smooth solvability of the initialvalue problem for the VlasovPoisson system with smooth initial data of unrestricted size, conditions under which the initialvalue problem for the VlasovMaxwell system has globalintime solutions (in both the smooth and weak senses), and more.
Audience
This book is intended for advanced graduate students and professionals with a high level of mathematical sophistication. A thorough graduatelevel study of partial differential equations and familiarity with advanced analysis are recommended.
Contents Preface; Chapter 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; Chapter 2: The Boltzmann Equation Near the Vacuum. Invariance of $xtv^2+xtu^2$, Sequences of Approximate Solutions, Satisfaction of the Beginning Condition, Proof that $u=\ell$, Remarks and Related Questions, References; Chapter 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; Chapter 4: The VlasovPoisson 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, Blowup in the Gravitational Case, References; Chapter 5: The VlasovMaxwell 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; Chapter 6: Dilute Collisionless Plasmas. The Smalldata Theorem, Outline of the Proof, Characteristics, The Particle Densities, Estimates on the Fields, Estimates on Derivatives of the Fields, References; Chapter 7: Velocity Averages: Weak Solutions to the VlasovMaxwell System. Sketch of the Problem, The Velocity Averaging Smoothing Effect, Convergence of the Current Density, Completion of the Proof, References; Chapter 8: Convergence of a Particle Method for the VlasovMaxwell System. Introduction, The Particle Simulation, The Field Errors, The Particle Errors, Summing the Errors, References; Index.
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1996 / xii + 241 pages / Softcover / ISBN13: 9780898713671 / ISBN10: 0898713676 / List Price $85.00 / SIAM Member Price $59.50 / Order Code OT52
