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The Cauchy problem in kinetic theory. (English) Zbl 0858.76001
Other Titles in Applied Mathematics. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xii, 241 p. (1996).
The last few years have seen the publication of several books in the field of nonlinear kinetic theory. The first of these was the little known book by N. Maslova [Nonlinear evolution equations (1993; Zbl 0846.76002)], and the second one was by C. Cercignani, R. Illner (this reviewer) and M. Pulvirenti [The mathematical theory of dilute gases (1994; Zbl 0813.76001)]. These books complement each other nicely. Maslowa’s text focuses on the Boltzmann equation, with special emphasis on the spatially homogeneous case, the linearized equation, and steady problems. Some little known results can be found there. The second mentioned book concentrates also on the Boltzmann equation, but there the emphasis is on rigorous validation, general existence and uniqueness theory, boundary conditions and particle simulation.
The book under review has little overlap with the other two books. This text presents a few selected topics related to the Boltzmann equation; after that, the second half of the book contains results on the Vlasov-Poisson and Vlasov-Maxwell equations (a field to which the author has made major contributions). The book seems very well suited for an advanced graduate course, and is indeed a compilation of lectures notes from such a course.
The first three chapters deal in great detail with the Boltzmann equation. After a brief formal derivation of the equation (a rigorous validation of the equation is not attempted), the basic properties are carefully introduced, and a global existence result for a rare gas cloud expanding in vacuum is given. Chapter 3, the longest chapter in the book, discusses global solutions of the Boltzmann equation near Maxwellian equilibria. The method used was pioneered by Kawashima and differs from the alternative one due to Ukai and Asano. Here is also a major difference in methodology relative to the second book mentioned above. The chapter makes for some though but rewarding reading for the patient; I know of no other text where the existence theory for the Boltzmann equation near equilibrium is written down in more detail, or more legibly.
The second part of the book deals with the Vlasov-Poisson and Vlasov-Maxwell systems. Chapter 4 presents in crystal clear fashion the global existence theorem for Vlasov-Poisson system as originally produced by Pfaffelmoser and simplified by Schaeffer. The chapter also contains blow-up results in dimension $$\geq 4$$ for the gravitational (stellar-dynamic) case, and for the relativistic Vlasov-Poisson system.
In chapter 4, a sufficient condition for the global existence of solutions to the relativistic Vlasov-Maxwell system is given; this condition amounts to global control of the growth of the “velocity” support of the solutions, an approach which is facilitated by the consideration of the relativistic case (“velocity” here is not the physical velocity; the physical velocity has $$c$$ as an upper bound). At first glance, the result seems paradoxical and almost vacuous, being a flavor: if the solution satisfies condition …, then it exists globally. The problem is that if you do not have a solution, you cannot talk about conditions it satisfies. What one faces here is a type of chicken and egg paradox. Fortunately, things are sound after all: what is really proved in this chapter is that a local solution can always be continued while the velocity support (of sequences of suitable linear approximations) remains compact.
This method is used in chapter 6, where it is shown that the velocity support does stay uniformly bounded if the given data satisfy a smallness condition; the calculations from chapter 5 then entail existence and uniqueness of global classical solutions, after a series of delicate estimates involving suitable weight functions.
Chapter 7 contains the global existence proof for weak solutions of Vlasov-Maxwell equations as pioneered by DiPerna and Lions, based on the velocity averaging method. Chapter 8 contains a convergence proof for a particle approximation in the “one and one-half” dimensional model (one spatial variable, but two momentum variables).
The book will make a very valuable addition to the bookshelf of every researcher in mathematical kinetic theory.
Portions of this review are reprinted with permission from SIAM Review, Vol. 39 (1997).

##### MSC:
 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C40 Kinetic theory of gases in time-dependent statistical mechanics 82C22 Interacting particle systems in time-dependent statistical mechanics