Hydrodynamic Limits of the Boltzmann Equation
The aim of this book is to present some mathematical results describing the transition from kinetic theory, and, more precisely, from the Boltzmann equation for perfect gases to hydrodynamics. Different fluid asymptotics will be investigated, starting alw
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1971
Laure Saint-Raymond
Hydrodynamic Limits of the Boltzmann Equation
123
Laure Saint-Raymond Département de Mathématiques et Applications Ecole Normale Supérieure 45 rue d’Ulm 75005 Paris France [email protected]
ISBN: 978-3-540-92846-1 DOI: 10.1007/978-3-540-92847-8
e-ISBN: 978-3-540-92847-8
Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008943981 Mathematics Subject Classification (2000): 76-02, 76P05, 82C40, 35B40, 35A99 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
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. The author is very grateful to Fabio Ancona and Stefano Bianchini for their invitation, and their encouragements to write these lecture notes. The aim of this book is to present some mathematical results describing the transition from kinetic theory, and more precisely from the Boltzmann equation for perfect gases to hydrodynamics. Different fluid asymptotics will be investigated, starting always from solutions of the Boltzmann equation which are only assumed to satisfy the estimates coming from physics, namely some bounds on mass, energy and entropy. In particular the present survey does not consider convergence results requiring further regularity. However, for the sake of completeness, we will give in the first chapter some rough statements and bibliographical references for these smooth asymptotics of the Boltzmann equation, as well as for the transition from Hamiltonian systems to hydrodynamics. Our starting point in the second chapter is some brief presentation of the Boltzmann equation, including its fundamental properties such as the formal conservations of mass, momentum and energy and the decay of entropy (for further details we refer to the book of Cercignani, Illner and Pulvirenti [31] or to the survey of Villani [106]). We then introduce the physical parameters characterizing the qualitative behaviour of the gas, and we derive formally the variou
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