Lecture Notes On Mathematical Theory Of The Boltzmann Equation

This is a collection of four lectures on some mathematical aspects related to the nonlinear Boltzmann equation. The following topics are dealt with: derivation of kinetic equations, qualitative analysis of the initial value problem, singular perturbation analysis towards the hydrodynamic limit and computational methods towards the solution of problems in fluid dynamics.

This book is based on the idea that Boltzmann-like modelling methods can be developed to design, with special attention to applied sciences, kinetic-type models which are called generalized kinetic models. In particular, these models appear in evolution equations for the statistical distribution over the physical state of each individual of a large population. The evolution is determined both by interactions among individuals and by external actions.Considering that generalized kinetic models can play an important role in dealing with several interesting systems in applied sciences, the book provides a unified presentation of this topic with direct reference to modelling, mathematical statement of problems, qualitative and computational analysis, and applications. Models reported and proposed in the book refer to several fields of natural, applied and technological sciences. In particular, the following classes of models are discussed: population dynamics and socio-economic behaviours, models of aggregation and fragmentation phenomena, models of biology and immunology, traffic flow models, models of mixtures and particles undergoing classic and dissipative interactions.

This book presents contributions on the following topics: discretization methods in the velocity and space, analysis of the conservation properties, asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and application of discrete models. It consists of ten chapters. Each chapter is written by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community. Contents: From the Boltzmann Equation to Discretized Kinetic Models (N Bellomo & R Gatignol); Discrete Velocity Models for Gas Mixtures (C Cercignani); Discrete Velocity Models with Multiple Collisions (R Gatignol); Discretization of the Boltzmann Equation and the Semicontinuous Model (L Preziosi & L Rondoni); Semi-continuous Extended Kinetic Theory (W Koller); Steady Kinetic Boundary Value Problems (H Babovsky et al.); Computational Methods and Fast Algorithms for the Boltzmann Equation (L Pareschi); Discrete Velocity Models and Dynamical Systems (A Bobylev & N Bernhoff); Numerical Method for the Compton Scattering Operator (C Buet & S Cordier); Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space (F Schrrer). Readership: Higher level postgraduates in applied mathematics.

This book is concerned with the methods of solving the nonlinear Boltz mann equation and of investigating its possibilities for describing some aerodynamic and physical problems. This monograph is a sequel to the book 'Numerical direct solutions of the kinetic Boltzmann equation' (in Russian) which was written with F. G. Tcheremissine and published by the Computing Center of the Russian Academy of Sciences some years ago. The main purposes of these two books are almost similar, namely, the study of nonequilibrium gas flows on the basis of direct integration of the kinetic equations. Nevertheless, there are some new aspects in the way this topic is treated in the present monograph. In particular, attention is paid to the advantages of the Boltzmann equation as a tool for considering nonequi librium, nonlinear processes. New fields of application of the Boltzmann equation are also described. Solutions of some problems are obtained with higher accuracy. Numerical procedures, such as parallel computing, are in vestigated for the first time. The structure and the contents of the present book have some com mon features with the monograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical problems. Because of this some chapters have been rewritten and checked again and some new chapters have been added.

"The material published in this volume comes essentially from a course given at the Conference on "Boltzmann equation and fluidodynamic limits", held in Trieste in June 2006." -- preface.

This book presents contributions on the following topics: discretization methods in the velocity and space, analysis of the conservation properties, asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and application of discrete models. It consists of ten chapters. Each chapter is written by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community. Contents:From the Boltzmann Equation to Discretized Kinetic Models (N Bellomo & R Gatignol)Discrete Velocity Models for Gas Mixtures (C Cercignani)Discrete Velocity Models with Multiple Collisions (R Gatignol)Discretization of the Boltzmann Equation and the Semicontinuous Model (L Preziosi & L Rondoni)Semi-continuous Extended Kinetic Theory (W Koller)Steady Kinetic Boundary Value Problems (H Babovsky et al.)Computational Methods and Fast Algorithms for the Boltzmann Equation (L Pareschi)Discrete Velocity Models and Dynamical Systems (A Bobylev & N Bernhoff)Numerical Method for the Compton Scattering Operator (C Buet & S Cordier)Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space (F Schürrer) Readership: Higher level postgraduates in applied mathematics. Keywords:Boltzmann Equation;Discrete Velocity Models;Evolution of a System of Equal Particles;Discretization Methods;Asymptotic Convergence;Initial/Boundary ConditionsReviews:“This book is a collection of high quality and very interesting articles dedicated to Henri Cabannes, one of the pioneers and prime movers of discrete kinetic theory, on the occasion of his 80th birthday … This is a really nice collection of articles and will be a very useful reference for some time to come.”Mathematical Reviews “This is a really nice collection of articles and will be a very useful reference for some time to come.”Zentralblatt MATH

In,1872, Boltzmann published a paper which for the first time provided a precise mathematical basis for a discussion of the approach to equilibrium. The paper dealt with the approach to equilibrium of a dilute gas and was based on an equation - the Boltzmann equation, as we call it now - for the velocity distribution function of such ~ gas. The Boltzmann equation still forms the basis of the kinetic theory of gases and has proved fruitful not only for the classical gases Boltzmann had in mind, but als- if properly generalized - for the electron gas in a solid and the excitation gas in a superfluid. Therefore it was felt by many of us that the Boltzmann equation was of sufficient interest, even today, to warrant a meeting, in which a review of its present status would be undertaken. Since Boltzmann had spent a good part of his life in Vienna, this city seemed to be a natural setting for such a meeting. The first day was devoted to historical lectures, since it was generally felt that apart from their general interest, they would furnish a good introduction to the subsequent scientific sessions. We are very much indebted to Dr. D.