Boundary Value Problems in Abstract Kinetic Theory

536 Pages·1987·10.06 MB·English

by  William Greenberg

OT23:
Operator Theory: Advances and Applications
Vol. 23
Editor:
I. Gohberg
Tel Aviv University
Ramat-Aviv, Israel
Editorial Office
School of Mathematical Sciences
Tel Aviv U Diversity
Ramat-Aviv, Israel
Editorial Board
A. Atzmon (Tel-Aviv) T. Kailath (Stanford)
J. A. Ball (Blacksburg) H. G. Kaper (Argonne)
K Clancey (Athens, USA) S. T. Kuroda (Tokyo)
L.A. Coburn (Buffalo) P. Lancaster (Calgary)
R G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) M. S. Livsic (Beer Sheva)
A. Dynin (Columbus) E. Meister (Darmstadt)
P. A. Fillmore (Halifax) B. Mityagin (Columbus)
C. Foias (Bloomington) J.D. Pincus (Stony Brook)
P. A. Fuhrmann (Beer Sheva) M. Rosenblum (Charlottesville)
S. Goldberg (College Park) J. Rovnyak (Charlottesville)
B. Gramsch (Mainz) D. E. Sarason (Berkeley)
J. A. Helton (La Jolla) H. Widom (Santa Cruz)
D. Herrero (Tempe) D. Xia (Nashville)
M.A. Kaashoek (Amsterdam)
Honorary and Advisory
Editorial Board
P. R Halmos (Bloomington) R Phillips (Stanford)
T. Kato (Berkeley) B. Sz.-Nagy (Szeged)
S. G. Mikhlin (Leningrad)
Springer Basel AG
W. Greenberg, C. van der Mee,
V. Protopopescu
Boundary Value Problems in
Abstract Kinetic Theory
1987
Springer Basel AG
Authors Adresses: Cornelis van der Mee
William Greenberg Department of Mathematics
Virginia Polytechnic Institute & Computer Science
& State University Clarkson University
BLACKSBURG, Va. 24061 POTSDAM, N.Y. 13676
U.S.A. U.S.A.
Vladimir Protopopescu
Engineering Physics & Matheinatics Division
Oak Ridge National Laboratory
OAK RIDG,E, Tennessee 37831
U.S.A.
Library of Congress Cataloging in Publication Data
Greenberg, William, 1941-
Boundary value problems in abstract kinetic theory
(Operator theory, advances and applications ; v. 23)
Bibliography: p
Includes index
1. Boundary value problems. 2. Initial value problems
3. Matter, Kinetic theory of. 1. Title. Il. Series.
III. van der Mee, Cornelis. IV. Protopopescu, Vladimir.
QA379.G735 1987 515.3'5 87-29981
ISBN 978-3-0348-5480-1 ISBN 978-3-0348-5478-8 (eBook)
DOI 10.1007/978-3-0348-5478-8
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Greenberg, William:
Boundary value problems in abstract kinetic
theory/William Greenberg; Cornelis van der Mee;
Vladimir Protopopescu. - Basel ; Boston ; Stuttgart
Birkhauser, 1987
(Operator theory ; Voi. 23)
NE: van der Mee, Cornelis:; Protopopescu, V.:; GT
Ali rights reserved.
No part of this publication may be reproduced, stored in a retrieval
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© 1987 Springer Basel AG
Originally published by Birkbăuser Basel in 1987.
Softcover reprint of the hardcover 1s t edition 1987
TABLE OF CONTENTS
Preface
Vlll
Chapter I
ELEMENTS OF LINEAR KINETIC THEORY
1. In traduction 1
2. Historical development 7
3. Semigroups 14
4. Positive cones and Banach lattices 18
Chapter II
STRICTLY DISSIPATIVE KINETIC MODELS
1. Introduction and historical development 23
2. Solutions in H 30
3. Bounded collision operators 42
4. Unbounded collision operators 48
Chapter III
CONSERVATIVE KINETIC MODELS
1. Preliminary decompositions and reductions 55
2. Boundary value problems 68
3. Evaporation models 76
4. Reflective boundary conditions 79
Chapter IV
NON-DISSIPATIVE AND NON-SYMMETRIC KINETIC MODELS
1. Indefinite inner product spaces 85
2. Reduction to a strictly dissipative kinetic model 92
3. Existence and uniqueness theory 97
4. Nonsymmetric collision operators 101
Chapter V
KINETIC EQUATIONS ON FINITE DOMAINS
1. Slab geometry 108
2. Boundary value problems for nonmultiplying slab media 109
3. Boundary value problems for multiplying slab media 119
4. Reflection and transmission operators 125
5. Slabs with reflective boundary conditions 131
vi
Chapter VI
EQUIVALENCE OF DIFFERENTIAL AND INTEGRAL FORMULATIONS
1. Kinetic equations in integral form 138
2. Preliminaries on convolution operators 140
3. Equivalence theorems 144
4. Reduction of dimensionality 152
5. Reflecting boundaries 154
6. Generalizations 157
Chapter VII
SEMIGROUP FACTORIZATION AND RECONSTRUCTION
1. Convolution operators on the half line 161
2. Semigroup reconstruction 168
3. Factorization of the symbol 179
4. Construction in a Banach space setting 189
5. Nonregularity of the collision operator 194
Chapter VIII
ALBEDO OPERATORS, H-EQUATIONS AND REPRESENTATION OF SOLUTIONS
1. Albedo operators and H-equations: the regular case 206
2. Albedo operators and H-equations: the singular case 213
3. Reflection and transmission operators and X- and Y -equations 217
4. Linear H-equations, uniqueness properties and constraints 228
5. Addition method 235
Chapter IX
APPLICATIONS OF THE STATIONARY THEORY
1. Radiative transfer without polarization 242
2. Radiative transfer with polarization 256
3. One speed neutron transport 274
4. Multigroup neutron transport 290
5. The Boltzmann equation and BGK equation in rarefied gas dynamics 302
6. A Boltzmann equation for phonon and electron transport 324
Chapter X
INDEFINITE STURM-LIOUVILLE PROBLEMS
1. Kinetic equations of Sturm-Liouville type 331
2. Half range solutions by eigenfunction expansion 339
3. Reduction to a modified Sturm-Liouville problem 342
4. Integral form of Sturm-Liouville diffusion problems and factorization 348
5. The Fokker-Planck equation 359
6. Electron scattering 362
vii
Chapter XI
TIME DEPENDENT KINETIC EQUATIONS: METHOD OF CHARACTERISTICS
1. In traduction 365
2. The functional formulation 369
3. Vector fields, function spaces and traces 374
4. Existence, uniqueness, dissipativity and positivity m Lp 384
5. The conservative case 391
6. Existence and uniqueness results m spaces of measures 395
Chapter XII
TIME DEPENDENT KINETIC EQUATIONS: SEMIGROUP APPROACH
1. Introduction and historical remarks 404
2. Existence, uniqueness, dissipativity and positiVIty in L 407
3. Connection between stationary and time dependent eq~ations 413
4. Spectral properties of positive semigroups 420
5. Spectral and compactness properties for kinetic models 432
Chapter XIII
APPLICATIONS OF THE INITIAL VALUE PROBLEM
1. Kinetic equations in neutron transport 440
2. Neutron transport (continued): spectral decomposition and hydrodynamics 452
3. Spencer-Lewis equation and electron deceleration 464
4. Electron drift in a weakly ionized gas 470
5. A transport equation describing growing cell populations 4 7 5
Bibliography 480
Subject Index 518
PREFACE
This monograph is intended to be a reasonably self -contained and fairly
complete exposition of rigorous results in abstract kinetic theory. Throughout, abstract
kinetic equations refer to (an abstract formulation of) equations which describe
transport of particles, momentum, energy, or, indeed, any transportable physical
quantity. These include the equations of traditional (neutron) transport theory,
radiative transfer, and rarefied gas dynamics, as well as a plethora of additional
applications in various areas of physics, chemistry, biology and engineering.
The mathematical problems addressed within the monograph deal with existence
and uniqueness of solutions of initial-boundary value problems, as well as questions of
positivity, continuity, growth, stability, explicit representation of solutions, and
equivalence of various formulations of the transport equations under consideration. The
reader is assumed to have a certain familiarity with elementary aspects of functional
analysis, especially basic semigroup theory, and an effort is made to outline any more
specialized topics as they are introduced.
Over the past several years there has been substantial progress in developing
an abstract mathematical framework for treating linear transport problems. The
benefits of such an abstract theory are twofold: (i) a mathematically rigorous basis has
been established for a variety of problems which were traditionally treated by
somewhat heuristic distribution theory methods; and (ii) the results obtained are
applicable to a great variety of disparate kinetic processes. Thus, numerous different
systems of integrodifferential equations which model a variety of kinetic processes are
themselves modelled by an abstract operator equation on a Hilbert (or Banach) space.
The ge11eral results so obtained are equally applicable to problems which range from
neutron transport in nuclear reactors and polarized light transfer through planetary
atmospheres, to reaction-diffusion processes m solutions, electron scattering in
semiconductors and metals, and cell growth in tumors. We attempt to present herein a
description of the methods and the history of this abstract g~neralization of kinetic
equations, at the same time discussing an extensive list of concrete physical applications.
The presentation divides into four subjects, which may be read somewhat
independently. We outline the division below.
ix
CHAPTER I I
CHAPTER I I I
Stationary problems
Differential methods
CHAPTER IV
CHAPTER v
CHAPTER VI
[
Stationary problems
CHAPTER VI I
Convolution methods
CHAPTER VI I I
I
CHAPTER XI
Time dependent problems
CHAPTER XI I
CHAPTER IX
Applications [ CHAPTER X
CHAPTER X I I I
There are some exceptions within the indicated divisions, however. Chapter X on
Sturm-Liouville equations develops aspects of the abstract theory. Section III.3 details
evaporation models. Section VIII.5 on adding methods is equally relevant to differential
and convolution formulations, although the method for representing solutions is certainly
integral. We have made a strenuous effort to separate the abstract theory from the
various applications. For example, Chapter IX, the longest chapter in the monograph,
consists entirely of examples of stationary transport problems, and is cross-referenced
throughout the text.
Four years ago, Birkhauser Verlag published a book [211] covering certain aspects
of linear kinetic theory. One may legitimately question whether the publication of a
second one is timely and opportune. We would point out that, in the last years, many
new - mostly abstract - results have been obtained which could not have been
contained m [211]. Specifically, Chapters IV, V, VII, VIII, X and XI, and parts of
Chapters III, VI, IX and XII are based on recent developments. The topics discussed in
Chapters II and III have largely been tackled in [211], but not in the generality and