The Cauchy Problem in Kinetic Theory

The Cauchy Problem in Kinetic Theory
Author: Robert T. Glassey
Publisher: SIAM
Total Pages: 254
Release: 1996-01-01
Genre: Science
ISBN: 9781611971477
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This 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.

The Cauchy Problem in Kinetic Theory

The Cauchy Problem in Kinetic Theory
Author: Robert T. Glassey
Publisher: SIAM
Total Pages: 246
Release: 1996-01-01
Genre: Science
ISBN: 0898713676
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Studies the basic equations of kinetic theory in all of space, and contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations. This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although describing very different phenomena, these equations share the same streaming term.

Mathematical Topics In Nonlinear Kinetic Theory

Mathematical Topics In Nonlinear Kinetic Theory
Author: Nicola Bellomo
Publisher: World Scientific
Total Pages: 245
Release: 1989-01-01
Genre: Mathematics
ISBN: 9814507482
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This book has the aim of dealing with the Nonlinear evolution problems related to the spatially dependent Boltzmann and Enskog equations.

Generalized Kinetic Models in Applied Sciences

Generalized Kinetic Models in Applied Sciences
Author: Luisa Arlotti
Publisher: World Scientific
Total Pages: 224
Release: 2003
Genre: Mathematics
ISBN: 9789812385604
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This book deals with analytic problems related to some developments and generalizations of the Boltzmann equation toward the modeling and qualitative analysis of large systems that are of interest in applied sciences. These generalizations are documented in the various surveys edited by Bellomo and Pulvirenti with reference to models of granular media, traffic flow, mathematical biology, communication networks, and coagulation models. The first generalization dealt with refers to the averaged Boltzmann equation, which is obtained by suitable averaging of the distribution function of the field particles into the action domain of the test particle. This model is further developed to describe equations with dissipative collisions and a class of models that are of interest in mathematical biology. In this latter case the state of the particles is defined not only by a mechanical variable but also by a biological microscopic state.

Mathematical Topics in Nonlinear Kinetic Theory II

Mathematical Topics in Nonlinear Kinetic Theory II
Author: N. Bellomo
Publisher: World Scientific
Total Pages: 228
Release: 1991
Genre: Science
ISBN: 9789810204471
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This book deals with the relevant mathematical aspects related to the kinetic equations for moderately dense gases with particular attention to the Enskog equation.

Advances in Kinetic Theory and Continuum Mechanics

Advances in Kinetic Theory and Continuum Mechanics
Author: Renee Gatignol
Publisher: Springer Science & Business Media
Total Pages: 307
Release: 2012-12-06
Genre: Science
ISBN: 3642502350
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This volume contains the proceedings of the symposium held on Friday 6 July 1990 at the University Pierre et Marie Curie (Paris VI), France, in honor of Professor Henri Cabannes on the occasion of his retirement. There were about one hundred participants from nine countries: Canada, France, Germany, Italy, Japan, Norway, Portugal, the Netherlands, and the USA. Many of his past students or his colleagues were among the participants. The twenty-six papers in this volume are written versions submitted by the authors and cover almost all the fields in which Professor Cabannes has actively worked for more than forty-five years. The papers are presented in four chapters: classical kinetic theory and fluid dynamics, discrete kinetic theory, applied fluid mechanics, and continuum mechanics. The editors would like to take this opportunity to thank the generous spon sors of the symposium: the University Pierre et Marie Curie, Commissariat a l'Energie Atomique (especially Academician R. Dautray and Dr. N. Camarcat) and Direction des Recherches et Etudes Techniques (especially Professor P. Lallemand). Many thanks are also due to all the participants for making the symposium a success. Finally, we thank Professor W. Beiglbock and his team at Springer-Verlag for producing this volume.

Kinetic Boltzmann, Vlasov and Related Equations

Kinetic Boltzmann, Vlasov and Related Equations
Author: Alexander Sinitsyn
Publisher: Elsevier
Total Pages: 321
Release: 2011-06-17
Genre: Mathematics
ISBN: 0123877806
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Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in 1938 and serves as a basis of plasma physics and describes large-scale processes and galaxies in astronomy, star wind theory. This book provides a comprehensive review of both equations and presents both classical and modern applications. In addition, it discusses several open problems of great importance. Reviews the whole field from the beginning to today Includes practical applications Provides classical and modern (semi-analytical) solutions

Many-Particle Dynamics and Kinetic Equations

Many-Particle Dynamics and Kinetic Equations
Author: C. Cercignani
Publisher: Springer Science & Business Media
Total Pages: 252
Release: 2012-12-06
Genre: Science
ISBN: 9401155585
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As our title suggests, there are two aspects in the subject of this book. The first is the mathematical investigation of the dynamics of infinite systems of in teracting particles and the description of the time evolution of their states. The second is the rigorous derivation of kinetic equations starting from the results of the aforementioned investigation. As is well known, statistical mechanics started in the last century with some papers written by Maxwell and Boltzmann. Although some of their statements seemed statistically obvious, we must prove that they do not contradict what me chanics predicts. In some cases, in particular for equilibrium states, it turns out that mechanics easily provides the required justification. However things are not so easy, if we take a step forward and consider a gas is not in equilibrium, as is, e.g., the case for air around a flying vehicle. Questions of this kind have been asked since the dawn of the kinetic theory of gases, especially when certain results appeared to lead to paradoxical conclu sions. Today this matter is rather well understood and a rigorous kinetic theory is emerging. The importance of these developments stems not only from the need of providing a careful foundation of such a basic physical theory, but also to exhibit a prototype of a mathematical construct central to the theory of non-equilibrium phenomena of macroscopic size.

Generalized Kinetic Models In Applied Sciences: Lecture Notes On Mathematical Problems

Generalized Kinetic Models In Applied Sciences: Lecture Notes On Mathematical Problems
Author: Luisa Arlotti
Publisher: World Scientific Publishing Company
Total Pages: 220
Release: 2003-08-12
Genre: Mathematics
ISBN: 9813106174
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This book deals with analytic problems related to some developments and generalizations of the Boltzmann equation toward the modeling and qualitative analysis of large systems that are of interest in applied sciences. These generalizations are documented in the various surveys edited by Bellomo and Pulvirenti with reference to models of granular media, traffic flow, mathematical biology, communication networks, and coagulation models.The above literature motivates applied mathematicians to study the Cauchy problem and to develop an asymptotic analysis for models regarded as developments of the Boltzmann equation. This book aims to initiate the research plan by the analyzing afore mentioned analysis problems.The first generalization dealt with refers to the averaged Boltzmann equation, which is obtained by suitable averaging of the distribution function of the field particles into the action domain of the test particle. This model is further developed to describe equations with dissipative collisions and a class of models that are of interest in mathematical biology. In this latter case the state of the particles is defined not only by a mechanical variable but also by a biological microscopic state.The book is essentially devoted to analytic aspects and deals with the analysis of the Cauchy problem and with the development of an asymptotic theory to obtain the macroscopic description from the mesoscopic one.