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The Cauchy Problem in Kinetic Theory

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by Robert T. Glassey


1996 / xii + 241 pages / Softcover / ISBN: 978-0-898713-67-1 / List Price $91.00 / SIAM Member Price $63.70 / 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, 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?

The topics treated include the study of the Boltzmann collision operator, the study of the initial-value problem for the Boltzmann equation with "small" and "near equilibrium" data, global smooth solvability of the initial-value problem for the Vlasov-Poisson system with smooth initial data of unrestricted size, conditions under which the initial-value problem for the Vlasov-Maxwell system has global-in-time solutions (in both the smooth and weak senses), and more.

This book is intended for advanced graduate students and professionals with a high level of mathematical sophistication. A thorough graduate-level study of partial differential equations and familiarity with advanced analysis are recommended.

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

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