
1996 / xii + 241 pages / Softcover / ISBN: 9780898713671 / List Price $85.00 / SIAM Member Price $59.50 / Order Code OT52
"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; Chapter 2: The Boltzmann Equation Near the Vacuum; Chapter 3: The Boltzmann Equation Near the Equilibrium; Chapter 4: The Vlasov–Poisson System; Chapter 5: The Vlasov–Maxwell System; Chapter 6: Dilute Collisionless Plasmas; Chapter 7: Velocity Averages: Weak Solutions to the Vlasov–Maxwell System; Chapter 8: Convergence of a Particle Method for the Vlasov–Maxwell System; References; Index.
ISBN: 9780898713671