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**Mathematical topics in nonlinear kinetic theory II. The Enskog equation.**
*(English)*
Zbl 0733.76061

Series on Advances in Mathematics for Applied Sciences 1. Singapore etc.: World Scientific (ISBN 981-02-0447-7/hbk; 981-02-0448-5/pbk). xii, 207 p. (1991).

This book may be regarded as the second part of a series of brief monographs devoted to various aspects of the kinetic theory of gases, the first being the work by N. Bellomo, A. Palczewski, G. Toscani [Mathematical topics in nonlinear kinetic theory (1988; Zbl 0702.76005)], to which the reader is referred for the knowledge of some relevant problems related to the Boltzmann equation. Indeed the Enskog equation is a model successfully attempted to take into account some effects of dense gases while the Boltzmann equation is well suited for rarefied gases.

The aim of this book is to provide the state-of-the-art of the analysis of the mathematical aspects initial value problems for the Enskog equation. The first chapter briefly describes the meaning of the standard Enskog equation and of the revised Enskog equation with its simplifications; it provides some definitions of solutions. The second chapter deals with initial value problem for the Enskog equation, in absence of external force fields, for small initial densities decaying to zero when the moduli of space and velocities variables go to infinite. Mild and classical solutions are proved to exist. The third chapter is devoted to the Cauchy problem for large (in the sense of \(L_ 1\) norm) initial data. The main idea is the use of a suitable Lyapunov functional and of a priori estimates to obtain the proof of existence of solutions. The fourth chapter deals with the asymptotic theory for the diameter of the spheres and/or the mean free path going to suitable limit values. The fifth chapter is devoted to comments and providds a list of open problems. Each chapter begins with a clear introduction and ends with an updated biliography. The book contains also an author index.

The aim of this book is to provide the state-of-the-art of the analysis of the mathematical aspects initial value problems for the Enskog equation. The first chapter briefly describes the meaning of the standard Enskog equation and of the revised Enskog equation with its simplifications; it provides some definitions of solutions. The second chapter deals with initial value problem for the Enskog equation, in absence of external force fields, for small initial densities decaying to zero when the moduli of space and velocities variables go to infinite. Mild and classical solutions are proved to exist. The third chapter is devoted to the Cauchy problem for large (in the sense of \(L_ 1\) norm) initial data. The main idea is the use of a suitable Lyapunov functional and of a priori estimates to obtain the proof of existence of solutions. The fourth chapter deals with the asymptotic theory for the diameter of the spheres and/or the mean free path going to suitable limit values. The fifth chapter is devoted to comments and providds a list of open problems. Each chapter begins with a clear introduction and ends with an updated biliography. The book contains also an author index.

Reviewer: G.Busoni

### MSC:

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

82B40 | Kinetic theory of gases in equilibrium statistical mechanics |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

82C40 | Kinetic theory of gases in time-dependent statistical mechanics |