Table Of ContentFM.qxd 8/12/05 1:33 AM Page i
S M M
pringer onographsin athematics
FM.qxd 8/12/05 1:33 AM Page iii
Jacek Banasiak Luisa Arlotti
Perturbations of
Positive Semigroups
with Applications
FM.qxd 8/12/05 1:33 AM Page iv
Jacek Banasiak, PHD, DSc Luisa Arlotti
School of Mathematical Sciences Department of Civil Engineering
University of KwaZulu-Natal University of Udine
Durban 4041 Via delle scienze 208
South Africa 33100 Udine
Italy
Mathematics Subject Classification (2000): 46-02, 47-02, 47D06, 46B42, 34G10, 47G20, 45K05, 46G10,
46N20, 46N30, 46N55, 46N60, 47N20, 47N30, 47N55, 47N60, 60J80, 82C40, 82D05, 82D37, 92D25,
47A55, 35B25
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2005929859
Springer Monographs in Mathematics ISSN 1439-7382
ISBN-10: 1-85233-993-4 e-ISBN 1-84628-153-9 Printed on acid-free paper
ISBN-13: 978-1-85233-993-
© Springer-Verlag London Limited 2006
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be repro-
duced, stored or transmitted, in any form or by any means, with the prior permission in writing of
the publishers, or in the case of reprographic reproduction in accordance with the terms of licences
issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms
should be sent to the publishers.
The use of 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 laws and regulations and there-
fore free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the infor-
mation contained in this book and cannot accept any legal responsibility or liability for any errors or
omissions that may be made.
Printed in the United States of America (MVY)
9 8 7 6 5 4 3 2 1
Springer Science+Business Media
springeronline.com
To our families
Contents
Preface ........................................................ xi
1 Introduction............................................... 1
1.1 What the Theory of Semigroups Is All About ............... 1
1.2 What This Book Is All About............................. 6
2 Basic Facts from Functional Analysis and Banach Lattices . 9
2.1 Spaces and Operators .................................... 9
2.1.1 General Notation.................................. 9
2.1.2 Operators ........................................ 13
2.1.3 Fundamental Theorems of Functional Analysis ........ 16
2.1.4 Adjoint Operators................................. 22
2.1.5 Vector-valued Functions and Bochner Integral......... 23
2.1.6 The Laplace Integral............................... 29
2.1.7 Vector-valued Analytic Functions and Resolvents ...... 33
2.1.8 Spaces of Type L.................................. 38
2.2 Banach Lattices and Positive Operators .................... 42
2.2.1 Defining Order.................................... 42
2.2.2 Banach Lattices................................... 50
2.2.3 Positive Operators................................. 52
2.2.4 Relation Between Order and Norm .................. 55
2.2.5 Complexification .................................. 60
2.2.6 Series of Positive Elements in Banach Lattices ........ 63
2.2.7 Spectral Radius of Positive Operators................ 65
3 An Overview of Semigroup Theory ........................ 69
3.1 Rudiments.............................................. 69
3.1.1 Definitions and Basic Properties..................... 69
3.1.2 Around the Hille–Yosida Theorem................... 72
3.1.3 Standard Examples................................ 74
3.1.4 Subspace Semigroups .............................. 77
viii Contents
3.1.5 Sobolev Towers ................................... 79
3.1.6 The Laplace Transform and the Growth Bounds of a
Semigroup........................................ 80
3.2 Dissipative Operators .................................... 83
3.2.1 Application: Diffusion Problems..................... 85
3.2.2 Contractive Semigroups with a Parameter ............ 92
3.3 Nonhomogeneous Problems ............................... 94
3.4 Positive Semigroups ..................................... 97
3.5 Pseudoresolvents and Approximation of Semigroups..........102
3.6 Uniqueness and Nonuniqueness............................108
4 Some Classical Perturbation Results.......................115
4.1 Preliminaries – A Spectral Criterion .......................116
4.2 Bounded Perturbation Theorem and Related Results.........121
4.3 Perturbations of Dissipative Operators .....................124
4.4 Miyadera Perturbations ..................................127
5 Positive Perturbations of Positive Semigroups .............133
5.1 Generalized Kato’s Perturbation Theorem ..................133
5.1.1 Resolvent Positive Operators........................141
5.2 Perturbation Results in L1 setting .........................143
5.2.1 Desch Perturbation Theorem .......................143
5.2.2 Kato’s Theorem in L1 setting .......................147
5.2.3 A Direct Proof of Corollary 5.17 ....................152
6 Substochastic Semigroups and Generator Characterization 157
6.1 Preliminaries............................................157
6.2 Strictly Substochastic Semigroups .........................159
6.3 Extension Techniques ....................................169
7 Applications to Birth-and-death Problems .................179
7.1 Preliminaries............................................179
7.2 Existence Results........................................180
7.3 Birth-and-death Problem – Preliminary Results .............183
7.4 Birth-and-death Problem – Substochastic Semigroup Approach185
7.4.1 Universality of Dishonesty ..........................190
7.5 Maximality of the Generator..............................191
7.6 Examples...............................................194
8 Applications to Pure Fragmentation Problems.............197
8.1 Preliminaries............................................197
8.1.1 Description of the Model ...........................198
8.1.2 Dishonesty and Nonuniqueness in Pure Fragmentation
Models...........................................200
8.2 Coefficient b(x|y) ........................................201
Contents ix
8.2.1 Power Law Case ..................................202
8.2.2 Homogeneous Case ................................203
8.2.3 Separable Case....................................204
8.3 Analysis of the Model....................................205
8.3.1 Well-posedness Results.............................205
8.3.2 An Approach Based on an Approximation Technique ..210
8.3.3 Full Description of Dynamics in the Separable Case....216
8.3.4 Uniqueness of solutions when K (cid:1)=Kmax .............224
9 Fragmentation with Growth and Decay....................229
9.1 Preliminaries............................................229
9.1.1 Description of the Models ..........................229
9.1.2 Dishonesty in Fragmentation with Decay and Growth..231
9.2 Fragmentation with Mass-loss.............................231
9.2.1 The Streaming Semigroup ..........................232
9.2.2 Well-posedness Results for the Full Semigroup ........242
9.2.3 Dishonesty .......................................252
9.2.4 Example .........................................260
9.2.5 Universality of Shattering ..........................261
9.2.6 Fragmentation Semigroup in the Finite Mass Space
L1([0,N],xdx). ...................................264
9.2.7 Fragmentation Semigroup in the Space L1(R+,dx) ....267
9.3 Fragmentation with Growth ..............................271
9.3.1 The Streaming Semigroup ..........................272
9.3.2 Back to the Growth-fragmentation Equation..........276
9.3.3 Dishonesty .......................................280
10 Applications to Kinetic Theory ............................285
10.1 Introduction ............................................285
10.1.1 General Definitions and Notation....................288
10.1.2 Linear Maxwell–Boltzmann Equation ................289
10.1.3 Linear Boltzmann Equation of Semiconductor Theory..291
10.2 Cauchy Problem for the Streaming Operator in Λ=R3×R3. .293
10.3 The Streaming Operator in Λ(cid:1)R3×R3 ...................300
10.3.1 Preliminaries .....................................300
10.3.2 The Maximal Free Streaming Operator and the
Existence of Traces ................................308
10.3.3 TheStreamingOperatorwithZeroBoundaryConditions311
10.4 Initial Boundary Value Problems for the Full Transport
Operator ...............................................320
10.4.1 Preliminaries .....................................321
10.4.2 Crucial Lemma ...................................322
10.4.3 Well-posedness of the Maxwell–Boltzmann Equation ...326
10.4.4 The Semiconductor Equation .......................330
10.5 Problems with General Boundary Condition ................352
x Contents
10.5.1 The Streaming Operator with Nonhomogeneous
Boundary Data ...................................352
10.5.2 The Streaming Operator with General Boundary
Conditions .......................................355
10.5.3 An Application to Multiplying Boundary Conditions...358
10.5.4 An Application to Conservative Boundary Conditions..362
11 Singularly Perturbed Inelastic Collision Models............371
11.1 Preliminaries............................................371
11.2 The Asymptotic Procedure ...............................372
11.3 The Model .............................................374
11.4 Mathematical Properties of the Collision Operators ..........378
11.4.1 Spaces and Operators..............................378
11.4.2 The Inelastic Collision Operator.....................379
11.4.3 The Elastic Collision Operator and Its Hydrodynamic
Space ............................................382
11.5 Well-posedness of the Problem ............................384
11.5.1 Model A .........................................386
11.5.2 Model B .........................................388
11.6 Asymptotic Analysis.....................................393
11.6.1 Derivation of the Scaled Equations ..................393
11.6.2 Limit Equations for Dominating Elastic Collisions .....394
11.6.3 Full Asymptotic Expansion .........................396
11.6.4 The Abstract Diffusion Operator ....................398
11.6.5 Solvability of the Kinetic-Diffusion Equation ..........402
11.6.6 Well-posedness in the Moment Spaces X .............408
k
11.6.7 Error Estimates ...................................416
11.6.8 Other Limit Equations .............................420
References.....................................................425
Index..........................................................435
Preface
Writing a preface is possibly the hardest part of work on any book. It is here
that the authors present, in a relatively lighthearted fashion, its content and
placeitinascientificandhistoricalcontextofthebroadfieldofknowledgeto
which it is relevant. So, let us start by explaining how this book came about,
what is in it, and why.
If one works for a long time on similar topics, then the results accumulate
and eventually reach a critical state in which the gaps left in theory are too
insignificant to justify separate papers but relevant enough not to be brushed
off with a notorious phrase: ‘It can be easily proved...’. At this stage one
can either move forward to explore new fields or rest for a while, playing
with the details of the theory. Quite often the choice is dictated by external
circumstances, as was the case with this book when one of the authors (J.B.)
was invited to spend two months as a Visiting Professor at the University of
Franche-Comt´einBesan¸conandhadtoprepareasetoflectures.Theideathat
materializeddatesbackseveralyearswhenJ.B.wasploddinghiswaythrough
the rich folklore of transport theory, trying to match rigourous mathematics
withphysicallyrelevantapplications.Whathereallyneededthenwasasingle
source that would combine mathematical tools of the trade with a guide to
how to use them in concrete models. This is an attempt to produce the book
that he would have liked to have had in his early days as a transport theorist
andwehopethatwehavesucceededinourendeavours.Thecompletionofthis
project, however, was possible only thanks to the expertise in kinetic theory
brought in by the second author (L.A.).
The book is intended to give a survey of relevant facts from functional
analysis, positivity theory, and theory of semigroups, presented at a not too
abstract but also not too superficial (we hope) level, together with many
proofs which are often difficult to find in the literature. On the other hand,
wediscussexamplescomingfromtheappliedsciences,frompopulationtheory,
throughfragmentationprocesses,tovariousaspectsoflineartransporttheory,
emphasisethedifferencebetweentheoriginalmodelanditsfunctionalanalytic
reformulation, which makes it tractable by techniques introduced in the first
xii Preface
part, and explore consequences of this dichotomy with major focus on the
phase transitions and existence of multiple solutions.
Choosing a reasonable level of presentation is a daunting task as every
single scientist has a unique mix of theory and applications with which he
or she feels at ease. For instance, a pure mathematician strives to achieve a
theorywhichisofultimategeneralityinapossiblymostconcisenotation,and
a scientist applying mathematics as a tool does not want to read hundreds
of pages of possibly beautiful but hermetic theory to get to a single piece of
information needed in a particular problem. The authors of this book belong,
in their opinion, to the realm of applied mathematics and thus, whilst trying
to uphold mathematical rigour, they sacrificed generality to present results
which are readily applicable and indeed, applied them to concrete physical
problems. On the other hand, the proofs may seem to be too detailed, but
it was the intention of the authors to write this book as an aid rather than
a challenge. Certainly, only the reader can judge to what extent the authors
managed to strike the desired balance of analysis and its applications.
Therearemanywaysofwritingabookdealingwiththesetopics.Wehave
chosen positivity as the main motive of our presentation. Without too much
exaggeration one can say that most of the ideas developed here were already
presentintheseminalpaperbyKato,[106].Itseems,however,that,especially
inanalysis,theywerelargelyforgotten,orusedonlyinanadhocfashion,until
the late 1980s when positivity reemerged in the theory of semigroups thanks
toresearchbyW.Arendt,C.Batty,R.Nagel,D.Robinson,andmanyothers,
andontheotherhanditwassystematicallyappliedinkinetictheorybypeople
such as J. Voigt, R. Beals, V. Protopopescu, C. van der Mee, and others.
Any book is a compromise between deadlines and our striving for perfec-
tion.Withoutdeadlinesnobookwouldbeeverpossible,asanyfinalversionis
butashadowofthePerfectBookexistinginourmindsandthus,bydefault,it
must be imperfect. This one is no exception, surely even more than we would
like to accept.
A number of results were obtained while writing the book and the proofs,
thoughtoourbestknowledgecorrect,didnothavetimetomatureandbecome
really elegant because of deadlines. We sincerely apologize for it and pray for
readers’ understanding.
Duetotimeandspaceconstraintswehaveleftoutmanyimportanttopics
(still, in the original contract the book was supposed to have 250 pages).
The most important absentees are: long-time behaviour of solutions, spectral
theory, compactness methods, and links with the probabilistic approach to
similar problems. The main reason for not including these topics is that the
authorsdonotfeelcompetentenoughtodiscussthematthelevelofaresearch
monograph. The first three are also well researched and easily accessible in a
numberofbooks(see,e.g.,[136,79,139,134,166]).Anothergreatabsenteeis,
of course, nonlinear theory of the presented problems, though in many cases
the results proved in the book form a necessary linear foundation on which
the nonlinear theory can be based.
