Mathematical Topics in Neutron Transport Theory I New Aspects •
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Series on Advances in Mathematics for Applied Sciences - Vol. 46
- - - - M Mokhtar-Kharroubi Laboratoire de Mathematiques et Mecanique Theorique Universite de Franche-Comte Besangon France
Mathematical Topics in Neutron Transport Theory
I
New Aspects
,IIIt World Scientific
Singapore· New Jersey· London· Hong Kong
•
Published by
World Scientific Publishing Co. Pte. Ltd . POBox 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Mokhtar-Kharroubi, M. Mathematical topics in neutron transport theory : new aspects I M. Mokhtar-Kharroubi. p. cm. -- (Series on advances in mathematics for applied sciences - Vol. 46) Includes bibliographical references and index. ISBN 9810228694 (alk. paper) I. Neutron transport theory -- Mathematics. 2. Mathematical physics. I. Title. II. Series. QC793.5.N4655M65 1997 621.48'31--dc21 97-21755 CIP
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In memory of my Father To Aurelia, Linda and Yacine
Nous ne vivons que pour decouvrir ia beautf-. Tout ie reste n'est qu 'attente. Khalil Gibran
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Acknowledgments
I am indebted to Nicola Bellomo for giving me the idea of writing this book, and for accepting it in the Series on Advances in Mathematics for Applied Sciences. I would also like to thank V. Caselles, M. Choulli, K. Latrach, K. Hamdache, J. Mazon and J. Voigt for their criticisms on various parts of this book and their encouragement. Writing this book gave me the opportunity to be familiar with mathematical typesetting ! Still, without the constant help of my colleagues B. Aoubiza, F. Ammar-Khodja and A. Benabdellah, I probably wouldn't have been able to finish this book in time. I am deeply grateful to them. I am also indebted to L. Fainsilber and C. Prequelin for helping me to improve the text in English. I finally wish to thank my analyst colleagues in Besanr;on, especially W. Arendt, Ph. Benilan, M. Choulli and R. Laydi, for so many years of fruitful contacts. [ cannot end this page without a thought for C. Bardos who initiated so many mathematicians of my generation into Transport Theory in the early nineteen eighties. [would like to pay homage to the role he had in the development of Kinetic Theory in France.
M.M-K Besanc;on, France November 1996
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And glancing back over the years he found that the one topic to which he has returned again and again is the problem of eigenvalues and eigenfunctions in its various ramifications. Hermann Weyl
Thus, the multiplication and criticality equations are characteristic value equations but their operators do not belong to the class for which the characteristic value theory is well established : they are not normal. Eugene P. Wigner
[ think the problem of constructing rigorous mathematical theories of criticality, in neutron chain reactors, will supply useful and interesting problems to mathematicians, for many years to come! / hope reactor physicists will regard the technical solution of such problems with due respect; in recent years, the phrase ''by physical intuition" has been too freely used as a substitute for "/ do not know why " ! Garrett Birkhoff
Concrete and functional analysis exist today in an inextricable symbiosis. When someone writes down a system of axioms, no one is going to take them seriously unless they arise from some intuitive body of concrete subject matter that you would really want to study and about which you really want to find out something. Felix E. Browder
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Contents 1
Introduction
1
2
Compactness properties of perturbed co-semigroups 2.1 Introduction.. . . . . . . . . . . . .. . . . 2.2 On essential type of co-seroigroups . . . . . . . . 2.3 On the strong convex compactness property . . . 2.4 Compactness properties of perturbed seroigroups 2.5 On the stability of the essential type 2.6 Comments References . . . . . . . . . . . . . . .
9 9 10 12 14 21 25 26
3
Regularity of velocity averages 3.1 Introduction . . .. . 3.2 Stationary problems 3.3 Evolution problems . 3.4 Comments . References .
29 29 31 39 44 46
4
Spectral analysis of transport equations. A unified theory 51 4.1 Introduction.. . . . . . . . . . . . . 51 4.2 Stationary problems . . . . . . .. 55 4.3 Evolution problems in lJ' (l
5
On the leading eigenelements of transport operators 5.1 Introduction.. .. . . .. . . . . .. .. 5.2 Spectral properties of positive operators Xl
99 99 100
CONTENTS
XlI
5.3 The irreducibility of transport semigroups 5.4 A general existence result 5.5 A spectral inequality 5.6 Nonexistence results . .. 5.7 Existence results . . . . . 5.8 Strict monotonicity properties of the leading eigenvalue 5.9 Domain derivative of the leading eigenvalue . . . . . . 5.10 An approximation theory of the leading eigenelements 5.11 The criticality eigenvalue problem 5.12 The effects of delayed neutrons 5.13 Comments. References. . . . . . . . . . .
108 112 114 116 118 124 128 130 133 136 139 141
6
Spectral theory of transport operators with form positive collision operators 145 6.1 Introduction.......... . . . . . . . 145 147 6.2 Self-adjointness and quadratic form . . . . . 6.3 Existence results for given spatial domains . 152 6.4 Existence results for large spatial domains 160 6.5 The isotropic models 163 168 6.6 Comments. References . . . . . . . . 170
7
On Miyadera perturbations of co-semigroups 7.1 Introduction.. . . . . . . . . . . . .. . 7.2 A perturbation theorem . . . . .. . .. . 7.3 The essential type of Miyadera perturbations 7.4 Comments. References . . . . .. . . . . . .
8
On resolvent positive operators and positive co-semigroups in U(f.£) spaces 185 8.1 Introduction... . . .. . . . . 185 8.2 A preliminary result . . . . . 186 8.3 Miyadera perturbations in L1(f.£) spaces 187 8.4 Alternative proof of Desch's theorem 191 8.5 Comments. 195 References . . . . . . . . . . . . . . . . . . 195
173 173 174 180 184 184
CONTENTS 9
rill
On singular neutron transport equations in Ll spaces 9.1 Introduction....... . . . . . . . . . . . . . 9.2 Generation results . . . . . . . . . . . . . . 9.3 The essential type of the perturbed sernigroup . 9.4 Comments . References . . . . . . . . . . . . . . . . .
10 Stochastic formulations of neutron transport. problems 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . .. . 10.2 Preliminary results. 10.3 Marimal solution of the stationary problem 10.4 The subcritical case . 10.5 The critical case . .. 10.6 The supercritical case 10.7 On the uniqueness . . 10.8 The time dependent problem 10.9 Time asymptotic behaviour 10.lOComments. References . 11 Velocity averages and inverse problems 11.1 Introduction . . . . . . . . . . . . . 11.2 One dimensional inverse problems 11.3 Multidimensional inverse problems 11.4 The dimension two .. . . . . . . . 11.5 Characterization of the range of integrated fluxes 11.6 Comments. References . .
197 197 199 203 213 214
Nonlinear 215 215 217 223 225 225 226 230 237 238 242 243 245 245 247 252 258 260 262 263
12 Limiting absorption principles and wave operators in U(Jl) spaces with applications to transport theory 267 12.1 Introduction . . . . . . 267 12.2 Preliminary results . . . . . 270 12.3 On the wave operators . . 275 12.4 The similarity of To and T 277 12.5 Converse results . . . . . . 278 12.6 Scattering theory for transport operators. 281 12.7 Comments . 286 References . . . . . . . . . . . . . . . . . . . . . 287
XlV
CONTENTS
13 Lin's factorization formalism and applications to transport theory 291 13.1 Introduction. . . . . . . . . . . . . . . . . . 291 13.2 A preliminary result . . . . . . . . . . . 292 13.3 On the wave operator s limt->+oo U(t)Uo( -t) 294 13.4 Application to transport groups. Part 1 297 13.5 Application to transport groups. Part 2 302 13.6 Comments . 306 References . . . . . . . . . . . . . . . . . . . . 307 14 Inverse scattering and albedo operator. By M. Choulli and P. Stefanov 309 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . 309 14.2 The special solution and the scattering operator . 314 14.3 Reconstruction of CYa , k from S . . . . . 320 326 14.4 The albedo operator . . 14.5 Reconstruction of CYa , k from $. The non-convex case. 330 References . . . . . . . . . . . . . . . .. . . .. .. . . 339 Index
343
Mathematical Topics in Neutron Transport Theory I New Aspects •
Chapter 1
Introduction This book is devoted to some recent mathematical developments in neutron transport theory. It covers several topics including spectral theory, scattering theory, nonlinear neutron transport eql:lations and inverse problems. Of course, those topics only deal with a few of the theoretical aspects of neutronic theory. However, they play an important part in the subject. The evolution of neutron transport theory is, to a large extent, related to the development of the nuclear industry since World War II. Nevertheless, as pointed out by I. Kuscer [27] and N.R. Corngold [16], it has older roots, in particular in the context of radiative transfer theory [15] . The field of neutronic theory, between applied mathematics and engineering, has benefited early from pioneering work by many physicists, engineers and applied mathematicians who gave the subject a great unity and maturity, handing down to the next generations a coherent body of problems and techniques. The references in some classical books [47] [7] and also in the proceedings of some of the first meetings on Transport Theory such as [1] [46], offer fair samples of the considerable literature which was written as early as the sixties. Later on, neutron transport theory flourished also within mathematics, shifted somewhat from practical probleins and developed as a branch of mathematics. This kind of evolution from physical problems to mathematical theory is a frequent phenomenon and is, in general, of great interest for mathematics even if its relevance to physics is a much debated question (see the nice introduction of R.D. Richtmyer's book [37]) . Because of the role of spectral problems in neutronic theory and thanks to pioneers such as M. Wing, J. Lehner, K. Jorgens, C .H. Pimbley, C . Birkhoff, V.S. Vladimirov, S. Ukai: and others, neutron transport theory already had strong connections, in the fifties, with sernigroup theory, spectral theory of non-self-adjoint operators, positive operators etc ... Thus, it is 1
2
Topics in Neutron Transport Theory
not surprising to see that neutron transport is related to functional analysis, as a lively field of applications and also as a good source for more abstract developments. A famous example is provided by Case's eigendistributions [13] [14] and the functional analytic tools and results they inspired [20] . Asymptotic phenomena in neutronic theory, as in many other fields, are of major interest. Three categories are worth mentioning. First, those related to various types of spectral problems: pulsed neutron experiments, diffusion length experiments or neutron wave experiments and the criticality problem in reactor theory [[19] Chap 5]. The second category deals with the connections between neutron transport equations and diffusion equations which are also used in nuclear reactor calculations. It turned out that the link between these so different equations is more rigorously described in terms of singular limits of properly rescaled transport equations, when a parameter (typically the mean free path) gets small [21] [25] [9] [4] [5]. The last category, namely scattering theory, which started in the seventies by H. Hejtmanek [22] and B. Simon [39], deals in particular with the asymptotic (t ---- 00) equivalence between free transport and transport with collisions. Generally speaking, neutron transport theory is mainly a perturbation theory with respect to free transport, and the perturbations (i.e. the collisions) are at the heart of the qualitative properties of neutron transport equations. This will appear throughout the different topics considered in the sequel. An important part of this book is devoted to mathematical developments around spectral and scattering theory. The equations we deal with are linear integra-differential equations, possibly with delayed neutrons. It is known that such equations provide an expected value theory of neutron transport. To describe the fluctuations from the mean behaviour, stochastic formulations for neutron chain fission were developed very early, in particular by L. Pal [34], G.!. Bell [6] and M. Otsuka and K. Saito [32] . We will also study such stochastic formulations in terms of probability generating functions. This formalism provides a more precise description of neutron chain fission. One interesting feature of the stochastic formulation of neutron transport is the nonlinearity of the relevant equations. The stochastic aspects must not be confused with those concerning linear equations with random data (cross-sections and sources) [36]. On the other hand, this class of nonlinear problems is not of the same nature as that occurring in neutron transport theory with feedback temperature [8] [35]. Lastly, we will consider inverse problems concerning the determination of source terms from partial informations on the solutions as well as inverse scattering problems. It is time to give a more precise idea of the content of the book. Chapter 2 is devoted to spectral theory of perturbed semigroups in Banach spaces. This classical theme was initiated by K. Jorgens [23] and 1. Vidav [40]
Chapter 1. Introduction
3
and developed by many others. The topic, tied to stability of essential spectra, is motivated by the understanding of the time asymptotic structure (t --) 00) of evolution transport problems. It is, of course, related to compactness problems. We will present some new functional analytic results on this topic. In particular, the use of the convex compactness property of the strong operator topology by L. Weis [45], J. Voigt [44] and G. Schliichtermann [38] makes for a very precise analysis of compactness problems in perturbation theory. This chapter is the functional-analytic background in preparation for Chapter 4 which is devoted to the spectral theory of transport semigroups or, more precisely, to a systematic analysis of various aspects of compactness which are of interest in the spectral theory of transport equations. Usually, the continuous and multigroup models are studied separately, partly because the latter are motivated by transport calculations but also for technical reasons. We will present a unified treatment of the compactness and therefore of the spectral analysis of neutron transport equations. The velocity space is endowed with an abstract (velocity) measure dJL(v) and thus the usual models are embedded into a unique formalism subject to suitable structure hypotheses. We show that the spectral theory of transport semigroups is essentially tied to the smoothness of iterated convolutions of certain measures related to dJL(v) . We will describe measures dJL( v) covering, of course, the usual models and ensuring such a regularity. Besides the generalization of known results, this yields a conceptual simplification of spectral theory of transport equations and a clarification of its natural scope. Compactness results by D. Aliprantis and O. Burkinshaw [2] and P. Doods and D.H. Fremlin [18], using domination arguments, are some of the main mathematical tools used in this chapter. This chapter is preceded, in Chapter 3, by a study of smoothing effects of velocity averages, in terms of compactness, which are also used in Chapter 4. Chapter 5 is devoted to the consequences of positivity in neutron transport theory, a topic which goes back to G. Birkhoff [10] [11] and I. Vidav [41]. This chapter is mainly devoted to a thorough analysis of the leading eigenelements. First we recall some classical results on the peripheral spectrum of positive semigroups in Banach lattices and the basic role of the concept of irreducibility. We will show how the combination of irreducibility and compactness yields, via B.de. Pagter's theorem [33] and superconvexity results by J.F.C. Kingmann [26] and T. Kato [24], very general existence results for eigenvalues. We will use some of I. Marek's results [29] [30] to prove the strict monotonicity of the leading eigenvalue with respect to the different parameters of transport equations and sketch an approximation theory of the leading eigenelements. Lastly we give a result on the "differentiability" with respect to spatial domain of the leading eigenvalue, for a model transport operator, and a simple formula for the
4
Topics in Neutron Transport Theory
derivative. We note that Positivity, which is at the heart of the peripheral spectral theory, is useless when we consider the non-peripheral spectrum which is one of the most challenging problems in neutron transport theory. Chapter 6 deals with the analysis of the real spectrum for transport operators involving positive (in the scalar product sense) collision operators. The general philosophy is that, as far as the real spectrum is concerned, the existence of a self-adjoint square root of the collision operator allows a symmetrization of the neutron eigenproblem and yields a very precise picture of the real point spectrum. The theory in question is purely Hilbertian and is independent of positivity in the lattice sense. The usual perturbation theory for neutron transport is based on standard tools for bounded perturbations. We present in Chapter 7 an extension of such tools to a class of unbounded perturbations introduced by 1. Miyadera [31] in the sixties and studied later, in particular by J. Voigt [42] [43]. We will show, in Chapter 8, how such tools are well suited to positive semigroups in Ll spaces because of the additivity of the norm on the positive cone. In Chapter 9, we show how this formalism yields a reasonable theory for singular neutron transport equations involving unbounded collision frequencies and unbounded collision operators motivated by free gas models. In Chapter 10, we consider a class of nonlinear transport problems arising in the stochastic formulation of neutron transport and involving non-local polynomial nonlinearities. We present a general mathematical analysis of such equations, which turns out to be strongly connected to spectral theory. Chapter 11 deals with a class of inverse problems consisting of the determination of source terms from velocity moments of the solution. By using an appropriate class of measures, we derive useful connections between the singularities of kernels of neutron integral equations and the fundamental solution of the Laplacian and show how to use them to recover explicitly the source terms. The last three chapters are devoted to scattering theory. In Chapter 12, we study the existence of wave operators for positive groups in Ll spaces and relate them to limiting absorption principles strongly tied to Positivity and to the Li framework. We also show how this formalism is particularly adapted to neutron transport equations. In Chapter 13, we deal with wave operators in reflexive Banach spaces by means of S.C. Lin's formalism based upon suitable factorization of perturbations, and show how transport equations in V' spaces (1 < p < 00) fit into this formalism. The last chapter, by M. Choulli and P. Stefanov, deals in particular with inverse scattering problems in transport theory. This book deals mostly with work by the author and his collaborators. However, an extensive bibliography, in each chapter, provides the reader with additional information on related publications. Moreover, each chapter ends with a section of comments which complements the content of
Chapter 1. Introduction
5
the chapter and where, sometimes, open problems, mostly of mathematical interest, are mentioned. No special prerequisite knowledge in transport theory is necessary for reading this book. Complete mathematical proofs are provided. However, we assume that the reader is familiar with some basic knowledge of functional analysis and spectral theory which are wellpresented, for instance in H. Brezis [12] or A.V. Balakrishnan [3] . The notions we need from semigroup theory are elementary and can be found in any textbook on the subject, for instance in E.B. Davies [17]. Some rudiments of measure theory are also useful in Chapters 3 and 4. On the other hand, the less classical results we need are explicitly stated and precisely referenced. A good general introduction to Transport Theory is given by J .J . Duderstadt and W.R. Martin [19] while neutronic theory is extensively covered by G.!. Bell and S. Glasstone [7] . The chapters of the present book are divided into sections. However, different theorems, propositions, ... , are only numbered according to the chapter where they are stated; for instance Theorem 7.4 refers to Theorem 4 in Chapter 7, while Remark 6.3 refers to Remark 3 in Chapter 6. Formulae are also numbered in a similar way. An index, at the end of the book, provides the reader with some key words. The reader interested primarily in spectral theory of transport equations may start with Chapter 4 and turn to the preceding chapters when necessary. Chapters 5 and 6, also devoted to spectral theory, are essentially self-contained. Before reading Chapter 9 on singular neutron transport equations, it is useful to read Chapter 8 and assume the main results of Chapter 7. The other chapters are independent. A word on the spirit of the book. While motivated by physical transport problems, this monograph is more functional-analysis oriented: whenever possible, the results are stated under the weakest possible assumptions, and the abstract results, independent of neutron transport theory, are emphasized and distinguished from those results which depend truly on the peculiarity of neutron transport equations even though the former are mostly motivated by the latter. The author hopes that this book will provide physicists and engineers with some useful mathematical tools and results. This book is also an attempt to stimulate mathematicians' interest in neutron transport theory. These words by E. P. Wigner [46] have always been with us: " The purpose of my remarks was to convey not only the impression that the science of reactors could be much helped by the attention of mathematicians interested in its problems, but also that mathematicians can find many interesting and challenging problems in reactor science" [1] !. Abu-Shumays et al (Ed) . Transport Theory. SIAM-AMS Proc. Providence. Rhode Island, 1969. [2] D. Aliprantis and O. Burkinshaw. Positive compact operators on
6
Topics in Neutron 'ITansport Theory
Banach lattices. Math. Z. 174 (1980) 289-298. [3] A.V. Balakrishnan. Applied Functional Analysis, Springer, 1976. [4] J. Banasiak and J.R Mika. Singularly perturbed evolution equations with applications to kinetic theory. Adv. Math. Appl. Sci. Vol 34, World Scientific, 1995. [5] C. Bardos, R Santos and R Sentis. Diffusion approximation and computation of the critical size. Trans. Amer. Math. Soc. 284(2) (1984) 617-649. [6] G.I. Bell. Stochastic theory of neutron transport. Nucl. Sci. Eng. 21 (1965) 390-401. [7] G.I. Bell and S. Glasstone. Nuclear Reactor Theory. Van Nostrand, 1970. [8] A. Belleni-Morante. Neutron transport with temperature feedback. Nucl. Sci. Eng. 59 (1976) 56-58. [9] A. Bensoussan, J.L. Lions and G.C. Papanicolaou. Boundary layers and homogenization of transport processes. J. Publ. RIMS. Kyoto Univ. 15 (1979) 53-157. [10] G. Birkhoff. Positivity and criticality. Proc. Symp. Appl. Math. XI. Amer. Math. Soc. Providence. RI. 1961. [11] G. Birkhoff. Reactor criticality in neutron transport theory. Rend. Math. 22 (1963) 102-126. [12] H. Brezis. Analyse Fonctionnelle:Theorie et Applications. Masson, Paris, 1983. [13] K.M. Case. Elementary solutions of the transport equation. Ann. Phys. 9 (1960) 1-23. [14] K.M. Case and P.F. Zweifel. Linear transport theory. Addison Wesley, 1967. [15] S. Chandrasekhar. Radiative Transfer. Dover, New York, 1950. [16] N.R Corngold. 50 years of neutron transport; in The Neutron and its Applications, Inst. Phys. Conf. Ser N064, Section 5, 1982. [17] E.B. Davies. One-parameter Semigroups. Academic Press, 1980.
Chapter 1. Introduction
7
[18] P. Doods and D.H. Fremlin. Compact operators in Banach lattices. Israel J. Math. 34 (1979) 287-320. [19] J.J . Duderstadt and W.R. Martin. Transport Theory. John Wiley & Sons, Inc, 1979. [20] W . Greenberg, C. Van der Mee and V. Protopopescu. Boundary Value Problems in Abstract Kinetic Theory. Birkhiiuser Verlag, 1987. [21] G.J . Habetler and B.J . Matkowsky. Uniform asymptotic in transport theory with small mean free path and the diffusion approximation. J. Math. Phys. 16(4) (1975) 846-854. [22] J. Hejtmanek. Scattering theory of the linear Boltzmann operator. Comm. Math. Phys. 43 (1975) 109-120. [23] K. Jorgens. An asymptotic expansion in the theory of neutron transport. Comm. Pure. Appl. Math. 11 (1958) 219-242. [24] T . Kato. Superconvexity of the spectral radius, and convexity of the spectral bound and the type. Math. Z. 180 (1982) 265-273. [25] J.B. Keller and E.W. Larsen. Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15(1) (1974) 75-81. [26] J.F.C. Kingman. A convexity property of positive matrices. Quart. 1. Math. Oxford. 12(2) (1961) 283-284. [27] I. Kuscer. A survey of neutron transport theory. Acta. Phys. A ustTiaca. Suppl. X (1973) 491-528. [28] S.C. Lin. Wave operators and similarity for generators of semigroups in Banach spaces. Trans. Amer. Math. Soc. 139 (1969) 469-494. [29] I. Marek. Frobenius theory of positive operators: Comparison theorems and applications. SIAM J. Appl. Math. 19(3) (1970) 607-628. [30] I. Marek. Approximation of the principal eigenelements in K-positive non-self-adjoint eigenvalue problem. Math. System. Theory. 5 (1971) 204-215. [31] I. Miyadera. On perturbation theory for semigroups of operators. Tohoku. Math. J. 18 (1966) 299-310. [32] M. Otsuka and K. Saito. Theory of statistical fluctuations in neutron distributions. J. Nucl. Sci. Tech. 2(8) (1965) 304-314.
8
Topics in Neutron Transport Theory
[33] B.de. Pagter. Irreducible compact operators. Math. Z. 192 (1986) 149-1.')3. [34] 1. Pal. Statistical theory of neutron chain reactions . Acta. Phys. Hung. 21 (1962), 390. [35] C.V. Pao. Comparison and stability of solutions for a neutron transport problem with temperature feedback. SIAM J. Math. Anal. 14(1) (1983) 167-184.
[36] G.C. Pomraning. Linear kinetic theory and particle transport in stochastic mixtures. Adv . Math. Appl. Sci. Vol 7, World Scientific, 1991. [37] R.D . Richtmyer . Principles of Advanced Mathematical Physics. Vol 1, Springer Verlag, 1978. [38] G. Schliichtermann . On weakly compact operators. Math. Ann. 292 (1992) 263-266. [39] B. Simon. Existence of the scattering matrix for the linearized Boltzmann equation. Comm. Math. Phys. 41 (1975) 99-108. [40] I. Vidav. Spectra of perturbed semigoups with applications to transport theory. J. Math. Anal. Appl. 30 (1970) 264-279. [41] I. Vidav. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22 (1968) 144-155. [42] J. Voigt. On the lJu"tnrhation theory for strongly continuous semigroups . Math. Ann. 229 (1977) 163-171. [43] J . Voigt . Stability of the essential type of strongly continuous semigroups. Trans. Steklov. Math. Inst. 203 (1994) 469-477. [44] J. Voigt . On the convex compactness property for the strong operator topology. Note di Mat. 12 (1992) 259-269. [45] L. Weis . A Generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129 (1988) 6-23. [46] E .P . Wigner et al (Ed) . Proc. Symp. Appl. Math. Xl. Amer . Math. Soc. Providence. R.1. 1961. [47] G.M . Wing. An introduction to transport theory. New York, Wiley, 1962.
Chapter 2
Compactness properties of perturbed co-semigroups 2.1
Introduction
The general form of neutron transport equations as integral perturbations of first order operators (streaming operators) suggests that their spectral analysis falls naturally within the framework of perturbation theory. Thus the following abstract setting was recognized very early as being well suited to neutron transport theory. We denote by X a complex Banach space, by {U (t) ; t ~ O} a co-semigroup in L (X) with generator T and by {V (t) ; t ~ O} the co-semigroup generated by A = T + B where BEL (X) . The operator T is intended to represent the streaming operator while B represents the collision operator. This chapter is devoted to the analysis of the asymptotic spectrum of A and V (t), in particular to understand the time asymptotic structure of solutions to Cauchy problems of the form
~~
= T
+ B
;
This classical functional analytic theme, strongly motivated by Transport Theory, was initiated by K. Jorgens [5] , 1. Vidav [22] and developed particularly by 1. Marek [7J, J . Voigt [23J [24J , Y. Shizuta [18J, L. Weis [27J , M. Mokhtar-Kharroubi [8J , F. Andreu, J . Martinez and J .M. Mazon [2J and G. Schliichtermann [14J. This chapter is devoted to some recent results on this topic in preparation for Chapter 4 where we deal with neutron transport equations. The questions are tied to the stability of essential spectra under suitable perturbations and are closely related to compactness prob-
9
Topics in Neutron Transport Theory
10
lems. We are mainly concerned with a very precise analysis of the different aspects of compactness which are of interest in the spectral theory of perturbed generators or perturbed semigroups. Two mathematical ingredients playa key role. Firstly, the Dyson-Phillips expansion of {V (t) ; t 2: O} and its remainders are interpreted as iterated convolutions of suitable strongly continuous operator-valued mappings and recursive fonnulae for them are derived. Secondly, we take full advantage of the convex compactness property for the strong operator topology by L. Weis [27] , J. Voigt [25] and G. Schliichtennann [15]. One of the main results is the stability of essential type if some remainder term is compact (or even strictly singular if X is isomorphic to some V(p,) space or to C(K) for a metric compact K) . We point out that in L1(p,) spaces the strictly singular operators coincide with the weakly compact ones (see A.Pelczynski [11]) and we have this framework in mind for applications to transport equations. Several other results are given. In particular, the compactness (resp. weak compactness, resp. strict singularity) of a remainder term is characterized in terms of known objects (the unperturbed sernigroup and the perturbation B) . We also show that the two sernigoups share many regularity properties, in particular the compactness properties. Furthermore, we clarify the link between the compactness properties of generators and those of sernigroups. Finally, we present a conjecture on partial spectral mapping theorems of interest for neutron transport operators in unbounded domains. The theory presented here will be the functional analytic background for a unified spectral analysis of neutron transport operators given in Chapter 4.
2.2
On essential type of co-semigroups
In this section we recall some properties of the essential spectral radius and related results. Let X be a complex Banach space and let 0 E L(X). Following J. Voigt [23], we define the "essential resolvent set" of 0 by
Pe(O) = p(O) U {>. E 0"(0); A is an isolated eigenvalue of finite multiplicity} where p( 0) is the resolvent set of O. We define the essential spectral radius of 0 by
re(O) = sup {IAI; Ai- Pe(O)} . There are connections between the essential spectral radius and the measure of noncompactness of 0 defined by
1I0lim = inf {1I0IMII;
M closed subspace of X with finite codimension}.
Chapter 2. Compactness properties of perturbed semigroups Then (see A. Lebow and M. Schechter [6]), norm on L(X) with the following properties
11811m =
11.lIm
11
is a multiplicative semi-
0 if and only if 8 is compact
and
We are now ready to present the concept of essential type for semigroups introduced by J. Voigt [23] and L. Weis [27]. Theorem 2.1 (L. Weis [27]). Let {U(t); t ;::: O} be a co-semigroup in L(X) . Then We
= lim C1log IIU(t)llm = inf {A E R;:3 t-->oo
M, IIU(t)llm ::; MeAt}
exists and (2.1) Definition 2.1 Let {U(t); t ;::: O} be a co-semigroup in L(X) with type w. The essential type of {U(t); t ;::: O} is We
E
[-oo,w]
defined by (2.1).
We close this section by explaining briefly how the essential type We is connected to the time asymptotic behaviour of the semigroup. Let T be its generator and W be its type. If We < w, then for each Q such that We < Q < w, aCT) n {.A, ReA;::: Q} consists of a finite and nonempty set of eigenvalues with finite algebraic multiplicities. Moreover, the spectral projection POI corresponding to {.A E aCT); ReA;::: Q} commutes with U(t) and there exists a constant COl such that IIU(t)(I - Pa)1I ::; Cae at . Thus the time asymptotic behaviour of the semigroup {U(t); t ;::: O} is determined by its part in the finite dimensional space Pa(X). Moreover, if X is a Banach lattice and U (t) ;::: 0 then
{.A
E
a(T); ReA;:::
W -
c:}
=
{w}
for € small enough and, when {U(t); t ;::: O} is irreducible, W is algebraically simple and the corresponding spectral projection is strictly positive. We refer to I. Marek [7], J . Voigt [26] and to Chapter 5 for details.
12
2.3
Topics in Neutron Transport Theory
On the strong convex compactness property
We recall some basic tools by L. Weis [27J, J. Voigt [25J and G. Schliichter mann [15], on the convex compactness property of the strong operator topology, we will use repeatedly in the sequel.
Theorem 2.2 (L. Weis [27J ,J. Voigt [25]). Let X, Y be two Banach spaces and let E c L(X, Y) be the subspace of compact operators. Let (0, p,) be a finite measure space and U :0
--+
E be bounded and strongly measurable
(i.e. U( .)x is measurable for all x E X). Then the strong integral x EX
--+
In
U(w)xdp,(w) E Y
is a compact operator.
Theorem 2.3 (G. Schliichtermann [15J) . Let X, Y be two Banach spaces and let E E L(X, Y) be the subspC\ce of weakly compact operators. Let (0, p,) be a finite measure space and U :0
--+
E be bounded and strongly measurable.
Then the strong integral x EX
--+
In
U(w)xdp,(w)
is a weakly compact operator.
The results above hold also for strictly singular operators in certain Banach spaces. Although it is not necessary for the sequel, we recall that a strictly singular operator 0 is defined by the fact that there is no infinite dimensional subspace on which the restriction of 0 is an isomorphism. The class of strictly singular operators contains the compact operators and coincides with them in Hilbert spaces. The "drawback" of this class is the lack of practical criteria in the LP(p,) spaces (1 < p < 00) P ~ 2.
Theorem 2.4 (L . Weis [27]) . Let X be a Banach space isomorphic to some LP(v) space (1 ::; p ::; 00) or to same C(K) with a metric compact
Chapter 2. Compactness properties of perturbed semigroups
13
K. Let E c L(X) be the space of strictly singular operators . Let (0, p,) be a finite measure space and U :0
---->
E be bounded and strongly measurable.
Then the strong integral x EX
---->
k
U(w)xdJ1(w)
is a strictly singular operator.
Finally, for the sake of completeness, we mention estimates of the measure of noncompactness of strong integrals, although we do not use them in the sequel.
Theorem 2.5 (L . Weis [27] , G. Schluchtermann and M. Kunze [16]) . Let X , Y be two Banach spaces, (0, J1) be a finite measure space and U : ,0
---->
L(X, Y) be bounded and strongly measurable.
i) If X ' is separable, then wE
0
---->
IIU(w)llm
is measurable
and
(2.2) ii) If X is separable, then
(2.3) iii) In general we have
(2.4) where
7 represents the upper integral.
We will comment on the results of this section at the end of this chapter.
Topics in Neutron Transport Theory
14
2.4
Compactness properties of perturbed semigroups
Let {U(t); t ~ O} be a co-semigroup on a complex Banach space ~ and let T be its generator. We denote by {V(t), t ~ O} the co-semigroup with generator A = T + B where B E L(X) is a perturbation. It is known that 00
V(t) =
L Uj(t),
(2.5)
o
where
Uo(t) = U(t) , Uj(t) =
lot U(t -
s)BUj_1(s)ds (j
~ 1)
(2.6)
and the series (2.5) converges in L(X) uniformly in bounded times. Moreover, the remainder term 00
R,,(t) =
L Uj(t) j=n
is given by
The following definition proves useful: Definition 2.2 Let f, 9 : [0, oo[
L(X) be strongly continuous. We define f (t - s )g( s )ds (strong integra0 . For repeated their convolution as f * 9 = convolutions, we also use the notation [fr = f * f * .. . * f (n times) with the convention [f] = f ·
J;
-t
We note that f * 9 is also strongly continuous and that f * 9 = 9 * f if f(s)g(t) = g(t)f(s) 'Vs, t E [0,00[. We start with a basic observation. Lemma 2.1 Uj(t)
= [UB]j * U
(j ~ 1) and R,,(t)
= [UBr * V
(n ~ 1).
Proof: By definition Uj = [U B] * Uj- 1(j ~ 1) so the first claim follows by induction. U(s)BV(t - s)ds = (UB) * V. We note that Rl(t) = V(t) - U(t) = Let n ~ 2 and
J;
15
Chapter 2. Compactness properties of perturbed semigroups Then the change of variables
=
It
. 0
Ln(un)V(t - un)du n = Ln * V,
where
Ln(un) =
l
un
dUn-I ··
·l
u 2
du 1 U(Ul)BU(U2 - Ul) ··· U(un - un-l)B .
On the other hand, Ln(u n ) is equal to
J
Un
Ln- 1 (Un-l)U(Un - un-l)Bdun-l = Ln- 1 * U B
o where
Ln-,(u,,_,)
~ [U[ du,,_, .. .
whence, by induction, Ln = L2
L 2(U2) =
l
f
1
du,U(u,)B · · · U(u n_, - un_,)B
* [U Bt- 2.
Finally,
u2
U(ul)BU(U2 - ul)Bdul = [U B]2
implies that Ln = [U Bt which ends the proof. 0 A useful consequence of Lemma 2.1 is the following recursive formula Lemma 2.2 The remainder terms of the Dyson-Phillips expansion (2.5) are related as Rn+! (t) = [U B] * Rn(t) (n;::: 1) . (2.8)
We note that (2.8) can be obtained more directly by observing that
Rn+!(t)
=",,00
~J=n+l
= rUB]
[UB]j *U= ""~
~J=n+l
* 2:;:n [UB]j * U =
We are now ready to prove
rUB]
[UB]
* [UB]j-l *U
* Rn(t) .
Topics in Neutron 'ITansport Theory
16
Theorem 2.6 A remainder term R",(t) (n 2: 1) is compact for all t 2: 0 (resp . weakly compact for all t 2: 0) if and only if Un(t) is compact for all t 2:0 (resp . weakly compact for all t 2: 0) . Proof: Let Un(t) be compact, then Un+1 (t) = J~ U(t - s)BUn(s)ds is compact by Theorem 2.2. It follows, by induction, that all Uj(t) (j 2: n) are compact and therefore R",(t) = I:':n Uj(t) is also compact since the series converges in L(X) . Conversely, let R",(t) be compact. Then, according to Lemma 2.2, R",+1(t) = rUB]
* R", =
lot U(t - s)BR",(s)ds
is compact by Theorem 2.2 and consequently Un(t) = R",(t) - R",+l(t) is also compact. The weak compactness is dealt with similarly. <> Remark 2.1 The result above remains true if we replace compact by strictly singular provided that the space X is isomorphic to some V(v) space (1 ::; p ::; 00) or to some C(K) space. It suffices to use Theorem 2·4· Remark 2.2 Since we are interested in giving compactness results in terms of known quantities (the unperturbed semigroup and the perturbation B) Theorem 2.6 shows that the relevant mathematical object is Un(t) = [UBt* U (or simply [UBt)(n 2: 1). Corollary 2.1 Let X be isomorphic to some Ll(v) space (or to C(K)). If R", (t) is weakly compact then R2n+1 (t) is compact. Proof: Let R",(t) be weakly compact. Then, according to Theorem 2.3, Un(t) = [U Bt * U is weakly compact so [U Bt+1is also weakly compact. On the other hand, using Lemma 2.2 n+ 1 times, R2n+1 = [UBt+1
* R", =
lot [UBt+
1
(t - s)R",(s)ds
and the term under the integral sign is compact as a product of two weakly compact operators in Ll(v) (or C(K))[4] whence R2n+l(t) is compact by Theorem 2.2. <> The continuity in time for the uniform topology of certain operators will play an important role in the sequel. To this end the following lemma proves useful. Lemma 2.3 Let f, g : [O,oo[ --+ L(X) be strongly continuous. If f or g depends continuously in t > 0 for the uniform topology then so does f * g . If g(t) is compact for t > 0 then f * 9 is continuous in t for the uniform topology.
Chapter 2. Compactness properties of perturbed semigroups
Proof Suppose that f : t > 0 uniform topology. We note that H(t) =
-+
17
f(t) E L(X) is continuous for the
lot f(t - s)g(s)ds
is obviously continuous at t = 0 since f ,9 : [O,oo[ -+ L(X) are locally bounded. Let to > O. We restrict ourselves to the continuity from the right at to . Then
H(to
.1: [f(to + h - s) - f(to - s)] g(s)ds 0
+ h) - H(to)
=
+[
to+h 0
f(to+h-s)g(s)ds=It+h
The second term 12 is clearly small in norm if h is small enough while the first one It decomposes as
J:
ex
O
[J(to + h - s) - f(to - s)] g(s)ds
-
I
+.
to to -ex
[J(to + h - s) - f(to - s)] g(s)ds .
One sees that second part of It is small (uniformly in h bounded) if a is small enough. Finally, fixing a > 0 small enough, the first part of It goes to zero in norm as h goes to zero because
f: [a, M]
-+
L(X) is uniformly continuous for finite M
and we are done. The arguments are the same if 9 (instead of f) is assumed to be continuous in t > O. Assume now that g(t) is compact for t > O. Then
It
=
+
J: J:
[J(to + h - s) - f(to - s)] g(s)ds [J(to
+h-
s) - f(to - s)] g(s)ds = J 1 + Jz.
The term Jz is clearly small in norm if a is small enough. Actually the difficulty lies in J 1 . We note that
to
11J1(X)II::; JOt 11[J(to+h-s)-f(to-s)]g(s)xllds.
Topics in Neutron Transport Theory
18 On the other hand,
s
sup II [J(to + h - s) - f(to - s)] g(s)xll
---+
I/xl/9
is lower semicontinuous and thus measurable. Hence
I!JIII::;
i
to
Ck
sup II[f(to+h-s)-f(to-s)]g(s)xllds . I/xl/9
We note that, for s fixed, {g(s)x; assumption so that
Ilxll ::; I}
is relatively compact in X by
sup II [J(to + h - s) - f(to - s)] g(s)xll
---+
0 as h
---+
0
I/xl/9
follows from the strong continuity of
i
to
Ck
f. Finally
sup 1I[J(to + h - s) - f(to - s)] g(s)xll ds
---+
0
ash---+O
I/xl/9
by the Lebesgue dominated convergence and this ends the proof of the second part. <> We can now characterize the continuity for the uniform topology of remainder terms. Theorem 2.7 A remainder term Rn(t) (n 2: 1) is continuous for the uniform topology in t 2: 0 if and only if Un(t) is. Proof. Let Un (.) be continuous in the uniform topology. Then, according to Lemma 2.3,
Un+1(t) =
lot U(s)BUn(t - s)ds
is continuous and, by induction, Uj (.) (j 2: n) are continuous. It follows that Rn(t) = Ej:n Uj(t) is also continuous for the uniform topology since the convergence is uniform in bounded times. Conversely, if Rn (.) is continuous in the uniform topology then, according to Lemma 2.2,
Rn+1(t) =
lot U(s)BRn(t - s)ds
is continuous from Lemma 2.3 and therefore Un(t) = Rn(t) - Rn+I(t) is also continuous. <> For the sequel we also need
Chapter 2. Compactness properties of perturbed seruigroups
19
Corollary 2.2 We assume that Rn(t) is compact. Then Un+l (.) is continuous for the uniform topology.
Proof Let Rn(t) be compact. Then, according to Lemma 2.3,
is continuous in the uniform topology and consequently Un+l(.) is also continuous from Theorem 2.7. <) We end this section with two results which clarify the link between compactness properties of perturbed generators and compactness properties of perturbed semigroups. Theorem 2.8 Let seT) be the spectral bound of T and let w be the type of the semigroup {U(t) ; t 2: O} . If Rn(t) is compact (resp . weakly compact) then T)-l B (A - T)-l
(p. -
r
is compact (resp.
weakly compact) for ReA> seT) . If Rn(t) is compact then, for every a > w, !!((A-T)-lB)n+l(A-T)-lL(x) -to as IImAI-too
(2.9)
uniformly in the region {A; ReA 2: a}. Proof Let Rn(t) be compact (resp. weakly compact). Then, according to Theorem 2.2 (resp. Theorem 2.3), Un = [U Bt * U is compact (resp. weakly compact) and consequently, according to Theorem 2.2 (resp. Theorem 2.3), the strong integral
foN e->.tUn(t)dt is compact (resp. weakly compact). On the other hand, for ReA> w,
foN e->.tUn(t)dt -t foOO e->.tUn(t)dt in L(X) as N -t so
10
00
00
(2.10)
e->.tUn(t)dt is compact (resp. weakly compact). Since the Laplace n transform of [U Bl * U is nothing but ((A - T)-l B) n (A - T)-l then the first part follows for ReA> wand, by analyticity arguments, for ReA>
20
Topics in Neutron Transport Theory
s(T) . To prove (2.9) we note, according to Corollary 2.2, that Un + 1 ( . ) is continuous in the uniform topology and therefore
uniformly in {>.; ReA ~ ex} by the RiemaIUl-Lebesgue Lemma. 0 We give now the converse of Theorem 2.8 in the context of Banach lattices and positive semigroups. To this end we recall the following definition Definition 2.3 A Banach lattice is said to have order continuous norm if any increasing net which has a supremum is convergent.
We point out that reflexive Banach spaces and L1 (J-L) spaces have order continuous norm ([10] p. 242) . Theorem 2.9 Let X be a Banach lattice such that X and X' have order continuous norm (resp. Let X = L 1 (f-L)) . Let {U(t); t ~ O} be a positive eo-semigroup and let B ~ O. We assume that Un (.) is continuous for the
uniform topology. If ((A - T) -1
B) n (A - T)
-1
is compact (resp. weakly
compact) then Rr,.(t) is also compact (resp. weakly compact). Proof: We start with
Let s > O. Then
and consequently J:+€ e->.tUn(t)dt is compact (resp. weakly compact) by domination [3] [1]. On the other hand,
because Un (.) is continuous for the uniform topology whence Un(s) is compact (resp. weakly compact). Finally Rr,.(s) is compact (resp. weakly compact) from Theorem 2.6. 0
Chapter 2. Compactness properties of perturbed semigroups
2.5
21
On the stability of the essential type
We recall that 00
V(t)
=
L Uj(t) j=O
where
Uj(t) =
lot U(t - s)BUj_1(s)ds (j 2: 1) , Uo(t) = U(t) .
Analogously, the seruigroup {U(t); t 2: O} may be considered as a perturbation of {V(t); t 2: O} 00
U(t) =
L Vj(t) j=O
where
Vj(t) =
lot V(t - s) (-B) Vj_l(s)ds (j 2: 1) , Vo(t) = V(t) .
We are going to show the symmetrical role of {U(t); t 2: O} and {V(t); t 2: O} as regards to certain properties.
Proposition 2.1 Let n E N be a non-negative integer. Then Un(t) is compact (resp . weakly compact) , (resp. strictly singular when X is isomorphic to some V(tt) space (1 :::; P :::; 00) or to C(K)) for t > 0 if and only if Vn(t) is compact (resp. weakly compact) , (resp . strictly singular) for t > O.
Proof We restrict ourselves to the compact case. The arguments for the other cases are exactly the same. We first treat the cases n = 0,1. Let Uo(t) be compact for t > O. Then Vo(t) = Uo(t)
+
lot Uo(t - s)BVo(s)ds
Topics in Neutron Transport Theory
22 is compact for t for t ~ O. Then
> 0 according to Theorem 2.2. Now let U1 (t) be compact
V1 (t)
===
J: J:
V(s)BV(t - s)ds
=-
Uo(s)BUo(t - s)ds -
-U1(t) -
J:
J: J:
[Uo(s) + R1(S)] B [Uo(t - s) + R1(t - s) ]ds Uo(s)BR1(t - s)ds -
Uo(s)BR1(t - s)ds -
J:
J:
R1(S)BV(t - s)ds
R1(S)BV(t - s)ds
so, according to Theorem 2.6, R 1(t) is compact and finally V1(t) is compact from Theorem 2.2. Let n ~ 2 and let Un(t) be compact. Then, according to Theorem 2.6, Rn(t) is compact. We note that
n-1
V(t) = Rn(t)
+L
Uj(t)
j=O
and
Vn
= [V (-B)t * V = (-It [VBt * V = (_I)n [RnB + 2:;':5 UjB]n * (Rn + 2:7:01 Ui )
,
i.e.
where C denotes a compact operator. For the sequel of the proof, we will denote by C every compact operator. We note that
Thus 00
Rn = [U Bt * U +
L
j=1
[U Bt+
j
*U
Chapter 2. Compactness properties of perturbed semigroups so that
00
linB
= [U Bt+l + L
On the other hand, Uj B
=
00
[U Bt+
j=l j ([U Bl * U) B
H1
=
j L [U Bl . j=n+1
= [U Blj+lwhence
Thus
and consequently, using Theorem 2.2, (2 .11) becomes
Moreover
n-1 n-1 n LUjB = UB+ L [UBlH1 = L[UBlj · j=O j=l j=l
Thus
Let Cj = [U Bl
v, ~ C + (_1)' ~ j
.
[;WBj;
r
,U,.
Then
i.e.
,
n. [U BjPI +2P2+·· + nPn ~ P1" " P' PI +"'+Pn=n' n· '"'
so that
23
Topics in Neutron Transport Theory
24 I.e.
Moreover, since PI
+ 2P2 + ... + npn + i k = (PI
~
n, letting
+ 2P2 + .. . + npn + i) -
n
shows that
is compact from Theorem 2.2 since Un is. Finally Vn is also compact. By reversing the role of U and V we show the converse result. <> It is easy to see that the above calculations combined with Lemma 2.2 show the following Proposition 2.2 Let n E N be a non-negative integer. Then Un(t) is continuous in t > 0 for the uniform topology if and only if Vn(t) is.
We are now ready to prove Theorem 2.10 Let some remainder term Rn(t) (n ~ 1) be compact (resp. strictly singular when X is isomorphic to some IJ' (p,) (1 :S p :S 00) or to C(K)). Then {U(t);t ~ O} and {V(t);t ~ O} have the same essential type.
Proof: According to Theorem 7.2, which holds even for certain unbounded perturbations, the essential type of {V(t); t ~ O} is less than or equal to the essential type of {U(t);t ~ O}. On the other hand, according to Theorem 2.6, the compactness (resp. the strict singularity) of Rn(t) is equivalent to that of Un(t), and hence to that of Vn(t) by virtue of Proposition 2.1. Thus the n-th remainder term of the Dyson-Phillips expansion of U(t), considered as a perturbation of V(t), is compact (resp. strictly singular) and, again by Theorem 7.2, the essential type of {U(t); t ~ O} is less than or equal to the essential type of {V (t ); t ~ O} . <> Remark 2.3 It follows from the general theory (see, for instance, [26]) that, for each a > We, (J" (T + B) n {.X; ReA ~ a} consists of .finitely many eigenvalues. It is possible to prove this result (actually a much weaker one) directly as is shown in the following. Proposition 2.3 Let W be the type of {U(t); t ~ O} and let some remainder term Rn(t) be compact. Then, for each a > w, (J" (T + B) n {A; ReA ~ a} consists of at most finitely many eigenvalues.
25
Chapter 2. Compactness properties of perturbed semigroups Proof: According to Theorem 2.8, [(A - T)-' B]
>
n+l
is compact and
uniformly in the half-space {A; ReA a). Then, according to Shrnulyan's theorem (see, for instance, [19] [12][21]), u (T B ) n {A; ReA > w) consists of (at most) isolated eigenvalues with finite algebraic multiplicity. On the other hand, it is easy to see that the A E up(T B ) n (ReA > w) n+2 if and only if 1 E up (A - T)-I B). Moreover [(A - T)-' B] <1 for IImAl large enough uniformly in {A; ReA 2 a}. Hence, for / ~ m ~ l " l a r ~ e enough, r, ((A - T)-' B) < 1 and therefore 1 $ up (A - T)-I B) . Finally, up (T B ) n {A; ReA 2 a) being discrete and confined in a compact set is necessarily finite. 0 We end this section with a plausible conjecture on a partial spectral mapping theorem. It is shown in Theorem 2.9, in the context of positive Q-semigroups on certain Banach lattices, that &(t) is compact if and only n if [(A - T)-' B] (A - T)-' is compact provided that (in(.) is continuous f o i t h e uniform'topology. Thus, the usual arguments for proving the discreteness of a (T B) n {A; ReA 1 we)
+
1
+
1
(
+
+
coincide with those ensuring the discreteness of
provided that some (in(.) is continuous for the uniform topology. This suggests the following Conjecture 2.1 Assume there exzsts n E N such that Un(.) is wntinuous for the uniform topology. Then the following partial spectral mapping theorem holds
2.6
Comments
This material in this chapter was taken &om [9]. The essential type was introduced by J. Voigt [23] who proved formula (2.1). An alternative a p preach which relies on the measure of noncompactness is due to L. Weis [27]
26
Topics in Neutron Transport Tbeory
who proved Theorem 2.1. Inequality (2.2) is due to L. Weiss [27] who used it, in particular, to prove the convex compactness property for the strong operator topology (Theorem 2.2) . However, L. Weis's proof contains a gap as was shown by J. Voigt [25J. An alternative proof of Theorem 2.2 which does not rely on formula (2.2) was given by J. Voigt [25] . It turned out that formula (2.2) is true provided that X' is separable. This was given by G. Schliichtermann and M. Kunze [16] as well as formulae (2.3) and (2.4) . The convex property for the strong operator topology for strictly singular operators, when X is isomorphic to V(p,) spaces or to C(K) (Theorem 2.4), is due to L. Weis [27]. For further informations on this topic we refer to [17] [25] . The stability result for essential type (Theorem 2.10) generalizes that of L. Weis [27], J. Voigt [24] and G. Schliichtermann ([17J Chapter 4 or [14]) where stronger assumptions are made. An inequality between the essential types is proved by J . Voigt [24J under more general assumptions on the remainder terms Rn(t) and for a class of unbounded perturbations B known as Miyadera perturbations. We will present J. Voigt's work [24J in Chapter 7 in preparation for Chapter 9 where we deal with singular neutron transport equations. Theorem 2.9 extends previous compactness results on positive semigroups on Banach lattices by M. Mokhtar-Kharroubi [8]. The conjecture above was stated in [8] under the loose assumption that U(t) is K-continuous in some sense and was answered positively by F . Andreu, J. Martinez and J .M. Mazon [2] under a stronger assumption than the one above. Finally, we mention a recent approach of the essential type of perturbed semigroups in Hilbert spaces, by Song Degong [20], based only on properties of the resolvent of the generator, in particular on its behaviour for large ImA.
References [1] D. Aliprantis and O. Burkinshaw. Positive compact operators on Banach lattices. Math. Z. 174 (1980) 289-298. [2] F. Andreu, J . Martinez and J.M. Mazon. A spectral mapping theorem for perturbed strongly continuous semigroups. Math. Ann. 291 (1991) 453-462. [3] P. Doods and D.H. Fremlin. Compact operators in Banach lattices. Israel J. Math. 34 (1979) 287-320. [4] N. Dunford and J .T. Schwartz. Linear Operators, Part 1. Interscience Publ,1958.
Chapter 2. Compactness properties of perturbed semigroups
27
[5] K. Jorgens. An asymptotic expansion in the theory of neutron t r a n s port. Comm. Pure. Appl. Math. 11 (1958) 219-242. [6] A. Lebow and M. Schechter. Semigroups of operators and measures of noncompactness. J. Funct. Anal. 7 (1971) 1-26. [7] I. Marek. Fundamental decay mode and asymptotic behaviour of positive semigroups. Czech. Math. J. 30(105) (1980) 579-590. [8] M. Mokhtar-Kharroubi. Compactness properties for positive semigroups on Banach lattices and applications. Houston J. Math. 1 7 NO1(1991) 25-38. [9] M. Mokhtar-Kharroubi. Spectra of perturbed semigroups in Banach spaces. Publications mathematiques de Besan~on,1995-1996.
[lo] R.
Nagel (Ed). One-parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, 1184, Springer Verlag, 1986.
[ll]A. Pelczynski. Strictly singular and cosingular operators. Bull. Acad. Sci. Polon. 13 (1965) 31-41.
[12] M. Ribaric and I. Vidav. Analytic properties of the inverse A(z)-' of an analytic linear operator valued function A ( z ) . Arch. Rational Mech. Anal. 32 (1969) 298-310. [13] M. Schechter. Quantities related to strictly singular operators. Ind. Univ. Math. J. 21 (11) (1972) 1061-1071. [14] G. Schliichtermann. Perturbation of linear semigroups. Preprint (1995). [15] G. Schluchtermann. On weakly compact operators. Math. Ann. 292 (1992) 263-266. [16] G. Schliichtermann and M. Kunze. Strongly generated Banach spaces and measures of non-compactness. To appear in Math. Nachr. [17] G. Schliichtermann. Properties of operator-valued functions and applications to Banach spaces and linear semigroups. Habilitationsschrift zur Erlangung der cenia legendi fiir das Fach Mathematik am Fachbereich Mathematik der Ludwig-Maximilians-Universitat1994. [18] Y. Shizuta. On the classical solutions of the Boltzmam equation. Comm. Pure. Appl. Math. 36 (1983) 705-754.
28
Topics in Neutron Transport Theory
[19J Y. Shmulyan. Completely continuous perturbation of operators. Dokl. Akad. Nauk SSSR 101 (1955) 35-38. English transl, Amer. Math. Soc. Transl. (2) 10 (1958) 341-344. [20J Song Degong. Some notes on the spectral properties of co-semigroups generated by linear transport operators. Transp . Theory. Stat. Phys. 26(1&2) (1997) 233-242. [21J S. Steinberg. Meromorphic Families of Compact Operators. Arch. Rational Mech. Anal. 31 (1969) 372-379 [22J 1. Vidav. Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30 (1970) 264-279. [23J J . Voigt. A perturbation theorem for the essential spectral radius of strongly continuous semigroups. Mon. Math. 90 (1980) 153-161. [24J J. Voigt. Stability of the essential type of strongly continuous semigroups. Trans . Steklov. Math. Inst. 203 (1994) 469-477. [25J J. Voigt. On the convex compactness property for the strong operator topology. Note. di. Mat. 12 (1992) 259-269. [26J J. Voigt. Positivity in time dependent linear transport theory. Acta. Appl. Math. 2 (1984) 311-331. [27J L. Weis. A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129 (1988) 6-23.
Chapter 3
Regularity of velocity averages 3.1
Introduction
As shown in the preceding chapter, compactness is at the heart of the spectral theory of perturbed operators. In particular, in neutron transport theory, a great deal of work has been devoted to the analysis of the compactness of suitable operators with applications to spectral problems. The literature goes back to the fifties and sixties, with J. Lehner and M. Wing [26],K. Jorgens [25],V.S. Vladirnirov [39],M.L. Demeru and B. Montagnini [15], M. Borysiewicz and J. Mika [9] to cite only a few. Since then, many other papers have been published and the available results cover, to a large extent, the needs for the usual transport operators. We refer to Chapter 4 for more information on the compactness in neutron transport theory. The relevant compactness results rely on the collision operator, a n integral operator with respect t o velocities, which is at the heart of the smoothing effects. The problems considered so far concern either continuous models (Lebesrgue measure on open subsets of Rn) or multigroup models (surface Lebesgue measures on spheres) motivated by nuclear reactor calculations. One of the main goals of this book is to unify the spectral theory of transport operators by dealing with abstract velocity measures covering, in particular, the usual continuous and multigroup models. As we will see in Chapter 4, this yields a better understanding of the spectral structure of such equations by revealing the natural boundaries of its scope. This unification is based on both the unification of the relevant compactness properties, and the functional analytic results of Chapter 2. The present chapter presents some
30
Topics in Neutron ll-ansport Theory
preliminary results on the regularity (compactness) of velocity averages. The smoothing effects of velocity averages, in terms of Sobolev regularity, are among the main achievements of kinetic theory in the last decade with the important contributions to nonlinear kinetic equations by R.J. Diperna and P.L. Lions. We will give the main references on Sobolev regularity of velocity averages in the last section. Before explaining the content of the present chapter, we recall a result by F. Golse, P.L. Lions, B. Perthame and R. Sentis [21]. Let p be a bounded positive Radon measure on Rn satisfying the following condition: There exists c > 0 and 0 < y < 2 such that
Let X be a smooth convex bounded open subset of the surface measure on dX. We define the sets
Rn and let &(x) be
where n(x) is the outward normal at x E dX. Let
where 1 5 p < oo,
and lv.n(x)IM x ) d ~ ( v ) ) .
LP(F*) = Then, the averaging operator
continuously into the fractional Sobolev space WSJ'(X) maps W? (and W+) where s = if p = 2 and 0 < s < id(;, $)y if p # 2. A consequence of this result is that p E Wz
-+IRn
p(.,v)dp(v) E LP(X) is compact.
The aim of this chapter is to extend this compactness result in two directions:
31
Chapter 3. Regularity of velocity averages
(i) We show that it holds for all bounded measures dJ.L such that the hyperplanes have zero dJ.L-measure. (ii) The convexity assumption on X is removed for homogeneous boundary conditions (i.e. 'PW- = 0 or 'PIN = 0). We note that (i) is contained in the general theory of P. Gerard [18] . However, we give here a direct proof for transport equations. We also give corresponding results for time-dependent problems, although we do not use them in the sequel. This extension provides us.with a unified spectral theory of transport generators in V spaces (1 < p < 00) (Theorem 4.1) given in Chapter 4 where many other compactness results are given. We note that the spectral theory of perturbed operators relies on specific compactness results as is shown in Chapter 2 at the functional analytic level and more concretely in Chapter 4 in the context of transport theory. This extension also proves useful for a unified analysis of the diffusion approximation of neutron transport-like equations [35].
3.2
Stationary problems
Let J.L be a bounded positive Radon measure on Rn such that the hyperplanes have zero dJ.L measure, i.e. for each e E sn-l , J.L{v E Rn;v .e Let 1 ::; p <
00.
= O} = O.
(3.1)
We define £p(R~ x R~) = £p(R~ x R~;dxdJ.L(v))
and
For each 'P E V(R~ x R~) we set
cP =
r 'P( ., v)dJ.L(v) E £p(R~). JRn
We start with Theorem 3.1 Let 1 < p < 00 and let (3.1) be satisfied. If:::: is a bounded subset of WP(R~ x R~) and if c Rn is open and bounded then the set {CPln;'P E::::} is relatively compact in V(n ;dx).
n
Before giving a proof we need two pre-limmary results.
Topics in Neutron TI-ansport Theory
32
Lemma 3.1 Let f.1, be a bounded positive Radon measure on Rn. Then the following assertions are equivalent: (i) The measure f.1, satisfies (3 .1) (ii) lime-+o sUPeESn-1 f.1, {v E Rnj Iv.el < c} = O.
Proof: Since f.1, is bounded, (ii) implies (i). To prove the converse, let us first show that V R > 0 lim sup f.1, {v E Rnj Ivl :::; Rj Iv.el e-+°eES n- 1
< c} = 0
(3.2)
implies (ii). Indeed, let a > 0 be fixed arbitrarily and let R > 0 be such that a f.1, {v E Rnj Ivl > R} :::; 2' ' then
f.1, {v E Rnj Iv.el < c}
= f.1, {v
E Rnj Ivl :::; R, Iv.el
:::; ~ + f.1,{v E
< c} + f.1, {v
Rnj Ivl :::; R, Iv.el
E Rnj Ivl
> R, Iv.el < c}
< c}.
According to (3.2), a sup f.1, {v E Rnj Ivl :::; R, Iv.el < c} < -2 eESn-l
for c small enough
and consequently lim sup f.1,{v E Rnj Iv.el < c} eESn-l
=0
E-->O
which proves the claim. Thus it suffices to show that (i) implies (3.2). Suppose that (3.2) is not true, i.e.
:3 R > 0 and a> 0 j V c > 0, :3 e E E sn-l and f.1,{Vj Ivi :::; R, Iv.eel Then, there exists e E sn-l such that e, Note that
---t
ease
---t
< c}
i.e. Ivl :::; R, Iv.el :::; c + Rle - eel} ::)
{Vj
a.
0 (a subsequence).
Iv.e,1 < c and Ivl :::; R,* Iv.el :::; c + Rle - eel,
{Vj
~
Ivl :::; R, Iv.e,1 < c}
Chapter 3. Regularity of velocity averages
33
so that p.{v;
Ivl ::; R, Iv.el ::; c: + Rle - eel}? a
V c: > 0
and finally which contradicts (i). <:; We recall also the local compactness criterion in V spaces (see, for instance, [10] p. 72) . Lemma 3.2 Let 1 ::; p <
lim
00
r If(x + h) -
h--->OjRn
and 3 C V(Rn) be bounded. If f(x)IP dx = 0
uniformly in f E 3
then, for any bounded set w C Rn, 31w is relatively compact in V (w). Proof of Theorem 3.1: We first consider the case p
8cp 9 = v 8x
+ cp
= 2.
Let
, cp E 3 .
(3.3)
Then 9 lies in a bounded subset of L2(R~ x R~) and 'P lies in a bounded subset of L2(R~). Hence, according to Lemma 3.2, it suffices to prove that IICPh - 'P11£2(R;:) ---> 0 as h --->
where 'Ph(X)
= 'P(x + h).
I ;;' - ~II
°
uniformly in cP E 3
(3.4)
0 uniformly in cP E 3
(3.5)
This amounts to
£2(Rn)
--->
0 as h
--->
where CPh is the Fourier Transform of CPh . Now (3.3) shows that g(~,v) = [i~.v+1J
and consequently
On the other hand,
kn
11
-
;;'W =
ei€.h
~(~) . Therefore (3.5) amounts to
ei€.hI21~(~)12 di;, ---> 0 as h ---> 0 uniformly in cP E 3.
(3.7)
34
Topics in Neutron Transport Theory
We split the above integral according to I~I following upper estimate
::; A or
I~I
2: A and derive the
Knowing that
{
il€I?A
l~wl2 dl, ---- 0 as A ---- 00 uniformly in
i.e.
{ il€I?A
dl,1 ( .g(~,v)
iRn t~ .v + 1
dJ-L(v) 12 ----0 asA----oo
(3.9)
uniformly in 9 bounded in L2 . By Cauchy-Schwarz inequality, (3 .9) is estimated by
< Setting e =
J
dJ-L(v)
sup [ ] I€I?AR:: 1+(~ .v)2
fti and p =
2
(3.10)
Ilglb .
I~I we note that, for every
J
c > 0 and
J
dJ-L(v)
I~I 2: A,
dJ-L(v)
[1+p2(e .v)2] -le.vl
J
dJ-L(v) + le.vl?c [1 + p2 (e.v)2]
< J-L {v; le.vl < c} + 1 + ~2c2 II dJ-L II <
sup J-L{v;le.vl
eESn-1
1
(3.11)
Thus, if I~I 2: A
dJ-L(v)
1 [1 + R::
(~ .v)2 ]
::; sup
eES n - 1
J-L {Vi le.vl < c} + 1
1
+
A2
c
2IIdJ-Lll·
(3.12)
Chapter 3. Regularity of velocity averages
35
According to Lemma 3.1, the first term on the right hand side of (3.12) is arbitrarily small for E small enough. Then, fixing E small enough, the second term of (3.12) is arbitrarily small for A large enough. Finally
which ends the proof of the case p = 2. To treat the case p # 2 we proceed as follows. Let
S : g E LP(RZ x RE) + cp E Wp(Ri x RE) be the solution operator of the equation
Let M be the averaging operator
and R be the restriction operator
where R is a bounded open subset of Rg. Then
RMS : LP(RZ x RE) + U ( R ) is bounded for 1 5 p < ca and is compact for p = 2 in view of the previous analysis. It follows, by an interpolation argument, that R M S is also compact for 1 < p < CO. 0
Remark 3.1 Assumption (3.1) in Theorem 3.1 is optimal. This is shown in the general setting of P Gerard [18]. We give here, in the context of transport theory, a simple argument communicated to the author by P. Gerard: Let toE Sn-' be such that p {v; v.t0 = 0)
> 0 and fn(x, v) = gn(v)einx.tO~(z)
where gn(v) = X { W ; ~ U . E ~ ~ - < ~ , ~ ~ ~ ~ M ) A E Coo(Rn) compactly supported, A(x) = 1for 1x1 5 R,
36
Topics in Neutron Damport Theory
where M > 0 is such that p { v ;v.(-, = 0 , Ivl 5 M ) > 0 and R verifies that there exists c > 0 such that
We note that
-
fn(x) =
/
> 0. One
f n ( x , v)dp(v)= ~(z)e""'"e, = e"x.b c, if 1x1 1 R
where %
=
J g n ( v ) d ~ ( v=) A V I s M ~ { ~ ; ~ v . & l < & ) ~ p ( ~ )
= p { v ;lvl
5 M , Iv.Eol I :) L p { ~IvI; I M , lv401 = 0 ) .
-
-
-
Hence c, is bounded away from zero. Taking a subsequence if necessary, we may assume that cn ? # o as n m. Finally { f n ( . ) ;n E N convergent subsequence in L%,(Rn) because of einx.cO. To deal with boundary value problems, we proceed as in [21]by appealing to an extension operator. First of all, we recall the relevant functional spaces and some of their properties. Let R be a smooth open subset Rn and let
p(n x RE)= LP(Rx RE;d x d p ( ~ ) ) We note that the elements of W P ( R x RE) have traces on a R x RE lying in suitable weighted spaces (see [13][14]).Let
r- = {(x,V ) E d R x R r ; v.n(x) < 0 ) ,I?+
= { ( x ,v ) E a R x
Rc;v.n(x) > 0 )
and
E(R
The spaces WP(R x RE) and x RE) are endowed with their natural norms. We recall that (see [13][14])
6 ( R x RE) = { c p E WP;Jr+ Icp(x,v)IPIv.n(x)I do(x)dll(v)< m)
37
Chapter 3. Regularity of velocity averages Finally, we defme the closed subspaces of
=(a
x RE)
We are now ready to state
Theorem 3.2 Let (3.1) be satisfied and let M be the averaging operator (P E
LP(Rg x R,") -+ (P E Lp(Rg)
where 1 < p < co. Then (i) For all w cc R, M : WP(R x RE) + P ( w ) is wmpact (ii) M : W K t LP(R) is compact (iii) M : W{+ -+ P ( R ) is wmpact (iv) If R is convex then M : =(!I x RE) + LJ'(R) is compact. Proof: The notation w cc R means that w is o-pen and TS C R . (i) Let I) E D(R) (Cw function with compact support in R) be such that +lw = 1 and E c WP(R x RE) be bounded. Then
is a bounded subset of WP(Rg x RE). According to Theorem 3.1,
is relatively compact in Lp(R). In particular, {(PI,; (P E 3 ) is relatively compact in Lp ( w ). (iv) When R is convex, there exists a continuous extension operator (see
(211) E : =(R x RE) + W P ( R zx R,"). Let c =(R x RE) be bounded. Then E ( 3 ) C W P ( Ex R:) is bounded and, by Theorem 3.1,
{GIo ; (P E E)
is relatively compact in V ( R )
which ends the proof since E ( P=~(P. (ii) Let S be the solution operator in the whole space
E Lp(Rg x RE) -+ 9 =
Jo
m
e-'g(x
+ s v , v)ds E w P ( gx
38
Topics in Neutron Transport Theory
of the problem 8
+
Let n c Rn be bounded, R be the restriction operator V(Rn) Then, according to Theorem 3.1,
RMS:
LP(R~
x
R~)
We-
be bounded. Then Let 3 C V(n x R~) and
-t
V(n).
V(n) is compact.
(3.13)
{g = v~ +
is bounded in
-t
r(x 'v)
io
e-Sg(x-sv,v)ds = Sn(g)
where
s(x,v) = inf {s > 0; x - sv 1c n} and Sn is defined by the expression above. Let E' be the trivial extension operator to Rn, then it is easy to see that
MSn(g) 5: RMSE' 9 ; 9 ~ O. Hence, in view of (3.13), MSn is compact by domination [3] [22] . {iii} Let 3 C W6+ be bounded and 9 = -v~ +
= io
e-Sg(x + sv, v)ds
= Sn(g)
where
T(X, v)
= inf {s
> 0; x + sv 1c n}
and Sn is defined by the expression above. As in {ii}, it is easy to see that
where
We conclude by a domination argument [3] [22] as in {ii}.
0
39
Chapter 3. Regularity of velocity averages
3.3
Evolution problems
To deal with time-dependent problems we need a slightly stronger assumption on the measure dfJThe linear varieties of codimension one have zero dfJ--measure.
(3.14)
By linear varieties of co dimension one we mean translated hyperplanes. We start with Lemma 3.3 Assumption (3.14) is equivalent to
(3.15)
Proof: It is based on the following remark: let (e, e') E R x R n , e2
+ e'2 =
1. Then {v; e + e' .v
lated hyperplane {v; e".v
=).}
= o}
is an empty set if e'
where e"
=
tT
and),
=
= 0 or
a trans-
Pi =
v';~e2 ,
e E ]-1,1[. In view of the finiteness of fJ-, (3.15) implies trivially (3.14). Conversely, by using similar arguments to those of Lemma 3.1 one shows that (3.14) implies (3.15) . <> We introduce the functional space W P=
{ cp E
LP(Rt x
R~ x R~); ~~ + v : : E LP(Rt x R~ x R~) }
and the averaging operator M: cp E £P(Rt
X
R~ x R~) ~
(
JR~
cp(t, x, v)dfJ- E £P(Rt x
R~).
The following result is the time-dependent version of Theorem 3.1. Its proof is similar to that of Theorem 3.1 . However, for the reader's convenience, we give a complete proof. TheoreIll 3.3 Let 1 < p < 00, and let (3.14) be satisfied. Let:=: c WP be bounded. Then, for all bounded set n c R t x R';, {
Proof Let (3.16)
40
Topics in Neutron llansport Theory
Then g lives in a bounded subset of P(& x Rz x Rc). Using the fact that
D(& x Rg x RE) is dense in WP, we easily obtain
where S E L ( P ( & x R; x RE)) is the solution operator of (3.16). Let R be the restriction operator
The problem amounts to proving the compactness of RMS : P ( R t x Rg x RE) + LP(R) (1 < p < 00). Finally, as previously, it suffices to consider the case p = 2. The Fourier Transform of (3.16) with respect to (t, x) gives
and then
According to the local compactness criterion (Lemma 3.2), we have to prove that ll@(h,k)- @llL2(RtxR;)
+
o
as (h, k) + o uniformly in cp E E
(3.19)
where @(h,k)(t, x) = @(t- h, x - k). By Parseval's identity, this amounts to
uniformly in cp E Z. For (h, k) fixed, we split the integral according to I(T, J) I < A or ((7,<)I 2 A and derive the upper bound
I
2
+
/
1
1
2
sup 11- e'[hr+k-tl I ~ @ I I $ ( ~ ~4 ~ ~ ) $(r,J) drdt (3.21) I(r,€)I
A
41
Chapter 3. Regularity of velocity averages
which shows that it suffices that the second term of (3.21) goes to zero as A -+ 00 uniformly in
Finally, it remains to show that (3.22)
• E 2 ~ Lete= ~ande = ~. Thene +e =land,forl(T,OI2:A .,.' +I€I'
.,.'+I€I
Hence (3.22) follows by Lemma 3.3. <) We derive now a similar result for half spaces (i.e. t 2: 0) . To this end we need an extension lemma. For the simplicity of notations, we denote by V the space £p(R-t x R~ x R~). Let WP(R+ ~'t
X
R nx x Rn) v =
{In E £P. £!£.at + v£!£. ax E V} .,.-
,
and let WP (Rt x R'; x R~) be the space
{
42
Topics in Neutron Transport Theory
Lemma 3.4 (Extension operator) There exists E E L(WP(Rt x R~ x R~), WP(Rt x R~ x R~)) such that Ecp(t,x,v) = cp(t , x,v) for t
Proof: Let cp E WP(Rt+ according to the rwe
X R~
'IjJ(t, x
> 0.
°
x R~). For t <
we define 'IjJ(t, x, v)
+ tv, v) = cp(O, x, v),
(3.23)
i.e. 'IjJ(t, x , v)
= cp(O, x -
(3.24)
tv, v).
We note that
r
k:x~
1'IjJ(t,x,v)jPdxdJ.L(v)=
J~x~
r
Icp(O,x,v)jPdxdJ.L(v) (t
={
cp for t 'IjJ for t
> <
and
Then 'IjJ'
°°
satisfies v
and
IN'
EN' LP(R ax + 7ft E
t X
Rn
x X
Rn) v
Ilv~: + at Lp Ilv:~ + :IILP =
J~T dt!R: XR;J 1'IjJ'(t,x,v)I
PdxdJ.L(v)
{ = IIcplI~p(Rt x R: X R;J) + T Ilcp(O,., ·)lliP(R: XR;J) ::; c Ilcpll£Vp where T >
°
and c is a constant depending on T. Finally, if p(.) E and p( t) = for t ::; -1, then
Coo (R t ) , p( t) = 1 for t >
°
°
Ecp = p(t)'IjJ' (t, x, v)
provides us with a continuous extension operator. <>
43
Chapter 3. Regularity of velocity averages Corollary 3.1 Let 1
00
and let (3.14) be satisfied. Let::::
bounded. Then, for all bounded set
n' c
LP(Rt
c WP be
x R:;), {
is
relatively compact in LP(n'). Proof: Let E be the extension operator of Lemma 3.4. Then {E
E ::::}
c WP(Rt x
R~
x
R~)
is bounded and then {E
cp. 0
Remark 3.2 We note that if
8
ax
8t
then
+ fot e-(t-s)g(s, x -
(t - s)v, v)ds.
Thus we can define the solution operator S : LP(R~ x R~) x LP(Rt+ x R~ x R~) (gl, g2)
---+
e- t g1 (x - tv , v)
---+
LP(Rt+
X
+ fot e-(t-s) 92(S , x - (t -
R~ x R~)
s )v, v)ds.
Then, according to Corollary 3.1, RMS is compact where R is the restriction operator and M is the averaging operator. We end this section with a compactness result for bounded (not necessarily convex) spatial domains. Let nCR'; be smooth and bounded. As previously, we denote by LP the space
LP(Rt+ x
n x R~)
and define
WP(Rt+ x
n x R~) =
{
%r + v* E LP}
W6_ = { O} W6+ = {
E
WP;
where WP = WP(Rt+ x
n x R~)
E
LP(n x R~), O}
Then
44
Topics in Neutron Transport Tbeory
Corollary 3.2 Let 1 < p < 00 and let (3.14) be satisfied. Let:::: C W6_ be bounded. Then, for all T > 0, {0;
9
=
~~ + v ~: +
Then
e-t
+
J:
where s(x,v)
(3.25)
e-(t-s)g(s, x - (t - s)v, v)X(t - s < s(x , v))ds
= inf {s > O;x - sv in}. We set
= Sn(
where Sn is the solution operator (3.25). Let
E: p(n x R~) x P(Rt x n x R~) -; P(R~ x R~) x P(Rt x R~ x R~) be the extension operator by zero outside Sn (
n.
Then
n x R~ ;
9 ~ O.
Hence
MSn:S RMSE where R is the restriction operator to [0, TJ x f1. We know that RM S is compact (Remark 3.2) whence M Sn is also compact by a domination argument [3J [22J . <;
Remark 3.3 Corollary 3. 2 remains true if we replace
3.4
W6_
by
W6+ .
Comments
The material in this chapter was partly used in M. Mokhtar-Kharroubi [34J and in Chapter 4. The compactness result with nonhomogeneous boundary conditions (Theorem 3.2 part (iv)) seems to be open if n is not convex; the problem being the lack (to our knowledge) of an extension operator. Corollary 3.2 is probably true for nonhomogeneous conditions on JO, T[ x an x R~ provided that n is convex; we did not try to elaborate on this point. We
Chapter 3. Regularity of velocity averages
45
note that Theorem 3.1, for the Lebesgue measure, appeared in F. Golse, B. Perthame and R Sentis [20] with a similar proof. We also mention that another approach to the compactness of velocity averages with respect to the Lebesgue measure for homogeneous boundary conditions, which relies on the dissipativity of the advection operator, was given by V.S. Vladimirov [39] for stationary problems and developed by M. Borysiewicz and J. Mika [9], A. Palczewski [36] and M. Mokhtar-Kharroubi [30] [31] . This result is usually stated in the form of the compactness of K (.A - T) -1 where T is the advection operator and K is the collision operator (i.e. the averaging operator). It turns out that this approach has a much larger scope (as long as we deal with Lebesgue measures) and covers also time-dependent problems and nonhomogeneous boundary conditions (see K. Jarmouni-Idrissi and M. Mokhtar-Kharroubi [24]). We also mention some results on the regularity of velocity averages for conservative reflection laws at the boundary when the traces are not in the relevant functional spaces (see A. Palczewski [37]). It seems that the idea of using velocity averages, to deal with nonlinear kinetic problems, emerged in the eighties with F. Golse, B. Perthame and R Sentis [20] , C. Bardos, F. Golse, B. Perthame and R Sentis [7], in the context of radiative transport equations and with K. Hamdache [23] for Boltzmann equations. We mention the general theory of P. Gerard [18] on micro local defect measures which has a much larger scope and contains, as by-product, the compactness of velocity averages. The smoothing effect of velocity averages, in terms of H! Sobolev regularity on the torus is given in V.l. Agoshkov [2] while a systematic analysis of Sobolev regularity of velocity averages, for general velocity measures in V spaces (1 < p < 00), was given by F . Golse, P.L. Lions, B. Perthame and R Sentis [21] using Fourier analysis in L2 spaces and interpolation arguments. Weak compactness results in L1 spaces as well as useful counterexamples are also given in [21]. The results were extended to more general (pseudodifferential) operators by P . Gerard [17] and P. Gerard and F. Golse [19] using tools from microlocal analysis. Much more general results on Sobolev regularity of velocity averages for transport operators were obtained by RJ. Diperna, P.L. Lions, and Y. Meyer [16] (see also M. Bezard [8]). Furthermore, such results are shown to be optimal by P.L. Lions [27]. This extensive work around velocity averages is motivated, to a large extent, by the study of nonlinear kinetic equations for which we refer to P.L. Lions [28] [29] and references therein. The smoothing effects of velocity averages are also useful for the analysis of hydrodynamic limits of nonlinear kinetic equations for which we refer to C. Bardos, F. Golse, and D. Levermore [4] [5]. We mention regularity results of velocity averages (1 < p < 00) by M. Mokhtar-Kharroubi [32] , when the source terms are even and smooth with respect to velocities. Such results, which are of interest for neutron transport approximations [33] [11] )
W;
46
Topics in Neutron llansport Theory
are based on singular integral analysis of Calderon-Zygmund type. We note also that velocity averages induce interesting dispersive effects (with respect to time) which have been used in the eighties by C. Bardos and P. Degond [6] to deal with Vlasov-Poisson equations. For more recent results in this direction we refer to B. Perthame [38],F. Castella and B. Perthame [12] and references therein. We will see in Chapter 13, in the context of scattering theory, that dispersive effects also come into play. Finally, velocity averages turn out to play an unsuspected role in the context of inverse problems (see Chapter 11).
References [I] V.I. Agoshkov. Spaces of functions with differential-difference characteristics and smoothness of solutions of the transport equation. Sou. Math. Dokl. 29 (1984) 662-666. [2] V.I. Agoshkov. Functional spaces, solvability conditions of problems for transport equation and smoothness of solutions. Project (Moscow). [3] D. Aliprantis and 0. Burkinshaw. Positive compact operators on Banach lattices. Math. 2. 174 (1980) 289-298. [4] C. Bardos, F. Golse and D. Levermore. Fluid dynamic limits of kinetic equations I: Formal derivations. J. Stat. Phys. 63 (1991) 323-344. [5] C. Bardos, F. Golse and D. Levermore. Fluid dynamic limits of kinetic equations 11: Convergence proofs for the Boltzmann equation. Preprint (1991). [6] C. Bardos and P. Degond. Global existence for Vlasov-Poisson Equation in 3 space variables with smal initial data. Ann. Inst Henri Poincare'. Anal Non Liniaire. 2 (1985) 101-118. [7] C. Bardos, F. Golse, B. Perthame and R. Sentis. The nonaccretive Radiative Tkansfer Equations: Global Existence Results and Rosseland Approximation. J. Funct. Anal. 77 (1988) 434-460. [8] M. Bezard. Regularit6 LP precishe des moyennes dans les equations de transport. Bull. Soc. Math. fiance. 22 (1994) 29-76. [9] M. Borysiewicz and J. Mika. Time behaviour of thermal neutrons in moderating media. J. Math. Anal. Appl. 26 (1969) 461-478.
Chapter 3. Regularity of velocity averages
47
[10] H. Brezis. Analyse Fonctionnelle:Theorie et Applications. Masson, Paris, 1983. [11] V. Caselles and M. Mokhtar-Kharroubi. On the approximation of the leading eigenelements for a class of transport operators. Transp . Theory. Stat. Phys. 23(4) (1994) 501-516. [12] F. Castella and B. Perthame. Estimations de Strichartz pour les equations de transport cinetique. C.R. Acad. Sci. Paris. Ser 1. 322 (1996) 535-540. [13] M. Cessenat. Theoremes de trace V pour des espaces de fonctions de la neutronique. C.R. Acad. Sci. Paris. Ser 1. 299 (1984) 831-834. [14] M. Cessenat. Theoremes de trace pour les espaces de fonctions de la neutronique. C.R. Acad. Sci. Paris. Ser 1. 300 (1985) 89-92. [15] M.L. Demeru and B. Montagnini. Complete continuity of the free gas scattering operator in neutron thermalization theory. 1. Math. Anal. Appl. 12 (1965) 49-57. [16] R.J. Diperna, P.L. Lions and Y. Meyer. V regularity of velocity averages. Ann. Inst Henri Poincare. Anal Non Lineaire. 8 (1991) 271-288. [17] P. Gerard. Regularization by averaging for solutions of partial differential equations. [18] P. Gerard. Microlocal defect measures. 16(11), (1991), 1761-1794.
Comm. Part. Diff. Eq.
[19] P. Gerard and F. Golse. Averaging regularity results for PDEs under transversality assumptions. Comm. Pure. Appl. Math. 45 (1992) 1-26. [20] F. Golse, B. Perthame and R. Sentis. Un result at de compacite pour les equations de transport et applications au calcul de la limite de la valeur propre principale d' un operateur de transport. C.R. Acad. Sci. Paris. Ser 1. 301 (1985) 341-344. [21] F. Golse, P.L. Lions, B. Perthame and R. Sentis. Regularity of the moments of the solution of a transport equation. 1. Funct. Anal. 76 (1988) 110-125. [22] P. Doods and D.H. Fremlin. Compact operators in Banach lattices. Israel 1. Math. 34 (1979) 287-320.
48
Topics in Neutron Transport Theory
[23J K. Hamdache. Theoremes de compacite par compensation et applications en theorie cinetique des gaz. C.R. Acad. Sci. Paris. Ser 1. 302 (1986) 151-154. [24J K. Jarmouni-Idrissi and M. Mokhtar-Kharroubi. Dissipativity of transport operators and regularity of velocity averages. Work in preparation [25J K. Jorgens. An asymptotic expansion in the theory of neutron transport. Comm. Pure. Appl. Math. 11 (1958) 219-242. [26J J . Lehner and M. Wing. On the spectrum of an unsyrrunetric operator arising in the transport theory of neutrons. Comm. Pure. Appl. Math. 8 (1955) 217-234. [27J P.L. Lions. Regularite optimale des moyennes en vitesses. Acad. Sci. Paris. Ser 1. 320 (1995) 911-915.
C.R.
[28J P.L. Lions. Compactness in Boltzmann equation via Fourier integral operators and applications. Part I and II. J. Math. Kyoto Univ. 34(2) (1994) 1-61. [29J P.L. Lions. Compactness in Boltzmann equation via Fourier integral operators and applications. Part III. J. Math. Kyoto Univ. 34(3) (1994) 539-584. [30J M. Mokhtar-Kharroubi. La compacite dans la tMorie du transport des neutrons. C.R. Acad. Sci. Paris. Ser 1. 303 (1986) 617-619 . [31J M. Mokhtar-Kharroubi. Time asymptotic behaviour and compactness in transport theory. Eur. J. Mech. B/ Fluids. 11 n01 (1992) 39-68. [32J M. Mokhtar-Kharroubi. W1 ,p Regularity in transport theory. Math. Models. Methods. Appl. Sci. 1 n04 (1991) 477-499. [33J M. Mokhtar-Kharroubi. On the approximation of a class of transport operators. Transp . Theory Stat. Phys. 22(4) (1993) 561-570. [34J M. Mokhtar-Kharroubi. A unified treatment of the compactness in neutron transport theory with applications to spectral theory. Publications mathematiques de Besanr-on, 1995-1996. [35J M. Mokhtar-Kharroubi. On the diffusion approximation for neutrons transport. Work in preparation.
Chapter 3. Regularity of velocity averages
49
[36] A. Palczewski. Spectral properties of the space inhomogeneous linearized Boltzmann operator. Transp. Theory Stat. Phys. 13 (1984) 409-430. [37] A. Palczewski. Velocity averaging for boundary value problems. Nonlinear kinetic theory and Mathematical Aspects of Hyperbolic Systems. Ed V.Boffi., F.Bampi, G.Toscani. Series Adv. Math. Appl. Sci. Vol 9, 1992. World Scientific. [38] B. Perthame. Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Part. Diff. Eq. 21 (1996) 659686. [39] V.S. Vladimirov. Mathematical Problems in the One-velocity Theory of Particle Transport. Atomic Energy of Canada. Ltd Chalk River. Ont Report. AECL-1661(1963) .
Chapter 4
Spectral analysis of transport equations. A unified theory 4.1
Introduction
Let n be a smooth open subset of Rn and let dJ-L be a positive Radon measure on Rn such that dJ-L {O} = O. We denote by V the support of dJ-L and refer to V as the velocity space. In this chapter, we first consider neutron transport equations (without delayed neutrons) of the form
af af at + v. ax + a(x, v)f(x, v, t) = K f
(4.1)
subject to boundary and initial conditions
f(t , ., ')Ir _ = 0 , f(O, x, v)
= fo(x, v)
where
r _ = {(x,v)
E
an x V;v .n(x) < O}
and n(x) is the outward normal at x E an. Here, a(x , v) denotes the collision frequency at x for neutrons with velocity v, while K is a (linear) collision operator having the peculiarity of being local with respect to the space variable x E n. As long as the collision frequency and the collision operator are time independent, the semigroup theory turns out to be the best framework to link the time asymptotic behaviour (t -+ 00) of the solutions to (4.1) with the spectral theory of suitable operators. It is convenient 51
52
Topics in Neutron Damport Theory
to write (4.1) as an abstract Cauchy problem
in the functional space
X = LP(O x V; dxdp(v))
(1 5 p
< oo)
where T is the streaming operator
Tf = -v.- af - a(x, v)f (x, v) dx
with domain
af
f E Lp(Ox V); v.-E
ax
LP(R.x V), fir- = O
We will assume that the collision frequency is non-negative and bounded a(.,.) E Ly(O x V).
(4.3)
It is known (see, for instance, [49])that T generates an explicit co-semigroup
where s(x,v)=inf{s>O;z-sv$O) We note that, when defined on an appropriate domain, T is still a generator under much more general assumptions on the collision hequency [49] and, moreover, most of the results of this chapter remain true without the boundedness assumption on the collision frequency. Actually, since a(.,.) is an absorption term, it can be unbounded without affecting the results of this chapter. We refer to Chapter 9, devoted to singular transport equations, where the unboundedness of u(., .) is fully exploited. We exclude the case p = co because U(.) is not strongly continuous in Lw(O x V). Thus, if the collision operator K is bounded in LP(R x V) then T K generates a co-semigroup {V(t);t 2 0) and therefore the Cauchy problem (4.2) is well-posed. Note also that much more general linear kinetic equations are well-posed in the semigroup setting (see [ll]Chap. XII). We recall (see [50] Lemma 1.1 and [49])that the essential type of {U(t); t 0) is given by
+
v = -,%
{
ess.
inf t<s(z,w)
t-I
Jof
>
}
u(x - SV, V ) ~ S
Chapter 4. Spectral analysis. A unified theory
53
and, if n is bounded,
a(U(t)) = {p. ; 1p.1 ::; e7J t } { a(T) = {A; ReA::; 7J} . We observe that if 0 ¢. V (i.e. the velocities are bounded away from zero) and if n is bounded, then {U(t); t ~ O} is nilpotent (it vanishes for t > .-
Finally, we point out that explicit expressions of 7J are also available for smooth nonhomogeneous collision frequencies ([18J Theorem 12.9, p. 271). The physical collision operators, used in nuclear reactor theory for all types of moderators (gas, liquid or solid), are bounded operators (at least in L2) and most of them are of the form Kcp =
i
k(x , v ,v' )cp(x ,v')dp.(v')
(4.5)
where k( .,., .) is a measurable function, and they are compact with respect to velocities (M. Borysiewicz and J. Mika [5], B. Montagnini and M.L. Demeru [29J and J.J. Duderstadt and W .R. Martin [9J Chapter 3). On the other hand, for solid moderators with isotropic properties (polycrystals) , the collision operator is not compact with respect to velocities (M. Borysiewicz and J. Mika [5J and E.W. Larsen and P.L. Zweifel [21]) . We note that collision operators of the form (4.5) bring some compactness with respect to velocities which will appear in a more precise form in the subsequent analysis. The purpose of this chapter is not to analyse a specific model but rather to present a general spectral theory of such equations. In other words, we are interested in a systematic analysis of the underlying compactness problems. Mathematically speaking, neutron transport equations involve two main parameters: (i) A m easure dp. (with respect to velocities) which determines the model we consider: the Lebesgue measure on Rn (or some absolutely continuous measure with respect to it) for the continuous models, the surface Lebesgue measure on spheres for the multigroup models or finitely many Dirac masses for the discrete models. (ii) A collision operator K E L(£P(n x V); dxdp.) (local with respect to positions) which describes the physics of scattering and production of particles (fissions).
54
Topics in Neutron Transport Theory
On the other hand, since the general aspects of the spectrum of transport operators rely on suitable compactness assumptions, our aim here is to delimit the class of parameters (df-L, K) for which the spectral theory of transport operators falls under the general theory of Chapter 2. Thus we aim, first, at unifying and extending the known compactness results by means of a unique formalism. Indeed, we will show that the relevant compactness results rely only on some geometrical (or analytical) properties of the measure df-L and on some compactness properties of the collision operator K in LP(V; df-L). Secondly, we will show converse results thus delimiting the natural class of parameters (df-L, K) beyond which the spectral structure of such equations should certainly be very different from the usual one and will require new mathematical tools. Thus the usual models (or even combinations of them) are embedded into a unique model subject to structure conditions on (df-L, K). This axiomatization yields a conceptual simplification and clarification of the spectral theory of neutron transport operators and opens new perspectives and problems that are discussed at the end of this chapter. In Section 4.2, we study the asymptotic spectrum of the generator a(T + K) n {A; ReA> s(T)} where
s(T) = sup {ReA; A E a(T)} . The spectral theory relies on the compactness of some power of K (A - T) -1. We introduce a class of collision operators K referred to as regular operators (i.e., roughly speaking, compact with respect to velocities). Byapproximating K by finite rank operators and using the averaging Theorem 3.2 we show that K(A - T)-1 is compact in V(n x V) (1 < p < (0) if the velocity measure df-L( v) is such that the hyperplanes have zero df-L-measure. We also show, by using comparison arguments [2J [8], the optimality of the class of regular collision operators. In £l(n x V) spaces, the strategy is different. By using Radon-Nikodym arguments, we give a sufficient geometrical condition on the velocity measure df-L( v) ensuring the compactness of K(A - T)-l K for convex n (or, at least the weak compactness for nonconvex domains). In Section 4.3, we deal with the spectral theory of the transport semigroup in V(n x V) (1 < p < (0). More precisely, we study stability of essential type following the abstract theory of Chapter 2. The aim is to show the compactness of some remainder term R.n(t) (m :::: 1) of the Dyson-Phillips expansion of the transport semigroup from the collisionness transport semigroup. We show, for regular collision operators, that the remainder terms act, with respect to the space variable, essentially as convolution operators with suitable measures related to the velocity measure df-L(v). This enables us to show that R 2 (t) is compact if
55
Chapter 4. Spectral analysis. A unified theory
the Fourier transform of (the truncations of) dJ.L vanishes at infinity. We show also that R3 (t) is compact if the translated hyperplanes have zero dJ.L-measure. A general (abstract) condition on the velocity measure dJ.L(v), ensuring the compactness of Rm(t) (m ~ 1), is also given. By using comparison arguments [2] [8], we show that no remainder term Rm(t) (m ~ 1) is compact in LP(n x V) if the collision operator K is not power compact in LP(V; dJ.L(v)). Section 4.4 is devoted to Ll(n x V) spaces. The general strategy is similar to that in Section 4.3. However, as opposed to Section 4.3 where we use Fourier analysis in L2(n x V) spaces and interpolation arguments, the compactness (or weak compactness) results in £1(n x V) spaces are derived by Radon-Nikodym arguments provided that the velocity measure dJ.L( v) satisfies a suitable geometrical condition. In Section 4.5, we extend the spectral results to transport equations with delayed neutrons by exploiting the compactness results of the previous sections. Finally, in the last section we comment on compactness in neutron transport theory and give several open problems of physical and (or) mathematical interest.
4.2
Stationary problems
This section is devoted to the spectral analysis of the unbounded operator
Acp = Tcp + Kcp with domain
where
r_=
{(x, v) E
an x V; v .n(x) < O} , x
= LP(n x V; dxdJ.L)
(1::; p < 00).
For the time being, K E L(LP(n x V; dxdJ.L)) is an abstract linear operator local in x E n, so it can be viewed as a mapping K:
xE
n -+ K(x) E L(£P(V; dJ.L)) .
We assume that K is strongly measurable, i.e. x E
n -+ K(x)f E £p(V)
is measurable for any f E LP(V)
and bounded, i.e. esssup IIK(x)IIL(LP(V)) xEn
< 00.
(4.6)
56
Topics in Neutron Transport Theory
It follows easily that K defines a bounded operator on the space V(D. x V; dxdf..L) according to the rule
r.p E U(D. x V)
~
K(x)r.p(x) E U(D. x V)
(we identify V(D. x V) and V(D.; V(V))) and
IIKIIL(x) :S esssup IIK(x)IIL(LP(V))' xEn
Let
s(T)
= sup {ReA; A E a(T)}
be the spectral bound of T. It is known (see 1. Vidav [46]) that
a(T + K) n {A; ReA> s(T)} consists of (at most) isolated eigenvalues if there exists n E N (n that [(A - T)-l is compact in U(D. x V).
~
1) such
Kf
(4.7)
This is a consequence of the so-called Gohberg-Shrnulyan's theorem (see [38] [42]). Thus, we are faced with the problem of determining the conditions under which (4.7) is satisfied. This is the main concern ofthis section. To this end we introduce a useful class of collision operators referred to as regular operators. Definition 4.1 Let E A collision operator
c L(V(V)) be the subspace of compact operators.
K: xED.
~
is regular if K(x) E E a.e, xED.
K(x) E L(U(V))
~
K(x) E E is measurable and
{K(x); x E D.} is relatively compact in L(LP(V)) . The two last conditions express that the collision operator depends "smoothly" on the space variable. They are satisfied if, for instance, 0. is bounded and x E ~ K(x) E L(U(V))
n
is piecewise continuous. The interest of this class lies in the following Proposition 4.1 The subspace of collision operators with kernels of the form EiEI Qi(x)fi(V)gi(V' ) where Qi E LOO(D.) , Ii E V(V),gi E V' (V) (~ + = 1) (I finite) is dense in the class of regular collision operators.
-?
Chapter 4. Spectral analysis. A unified theory
57
Proof It follows by approximating K in LOO(nj E) by finitely valued functions and approximating their values by finite rank operators. <; The first result of this section is the following
Theorem 4.1 Let 1 < p < 00 and let n be bounded. We assume that dp. is such that the hyperplanes have zero dp.-measure and that the collision operator is regular. Then K(>. - T)-l and (>. - T)-l K are compact in V(n x Vj dxdp.) . Proof: According to Proposition 4.1 (and by linearity), it suffices to give a proof for a collision operator K with a kernel of the form k(x, v, v') = ex(x)f(v)g(v ' ) where I
ex E LOO(n), f E U(V), g E LP (V) .
By approximating f and g by continuous functions with compact supports, we may suppose, without loss of generality, that f and g are continuous with compact supports. In such a case, K(>. - T)-l and (>. - T)-l K map £T(n x Vj dxdp.) into itself for all r E [1 , 00] so that (by interpolation) it suffices to give a proof for p = 2. We consider first K(>. - T)-l. Let Mg be the averaging operator
It suffices to show that
This amounts to
Mg : D(T) = { 'P E X; v . ~: EX, 'Plr _ =
o}
-+
L2(n)
is compact.
By decomposing g into positive and negative parts, we may suppose that g is non-negative. Hence the result follows from Theorem 3.2. Similarly,
Mf : D(T*) = { 'P E Xj v . ~: EX, 'PW + =
o}
-+
and consequently
K*(5. - T*)-l is compact. Hence (>. - T)-l K is compact by duality.
<;
L2(n)
is compact
58
Topics in Neutron Transport Theory
Remark 4.1 As pointed out in Chapter 3 (Remark 3.1) the assumption on the measure df.L is optimal.
We prove now several results showing the optimality of the class of regular collision operators. Theorem 4.2 Let 1 < p < 00 and let n be bounded. We assume that V is bounded and that the collision operator is independent of x E n. If K(>..-T)-l is compact in lJ'(n x Vj dxdf.L) then K is compact in lJ'(Vj df.L).
Proof: Let K(>" - T)-l be compact on lJ'(n x Vj dxdj.t) (1 < p < 00). We recall that
(A\
-
T)-l cp =
l
S
(X'V)
u(x-rv,v)dr) ( e -(.>.s+.r 0 cp x - sv, v )d s
o so that its restriction to the closed subspace lJ'(Vj df.L) is the multiplication operator cp E LP(Vj df.L)
_l
where
uL1( x, v ) -
S
(X'V)
o
B(x, v)cp(v)
-7
e-('>'8+
f: 0
u(x-rv,v)dr)d
s.
Let Sen, S compact and let P be the operator cp E U(n x Vj dxdf.L)
-7
Pcp =
Is
cp(x, v)dx E U(Vj df.L).
Then PK(>"-T)iL~(V): U(Vjdf.L)
-7
U(Vjdf.L) is compact
(4.8)
and, since P and K commute, it follows that PK(>" - T)iL~(V) = KM
where M : lJ'(Vj df.L)
-7
lJ'(Vj df.L) is the multiplication operator by B(v)
=
Is
B(x, v)dx.
We note that M is invertible because B(x, v)
~
1
dist(Sj80) VO
e-('>'+lIulloo)Sds
j
xES
where Vo is the maximum speed. Hence K is also compact. <> We can also deal with nonhomogeneous collision operators as is shown in the following
59
Chapter 4. Spectral analysis. A unified theory
Theorem 4.3 Let V be bounded and the collision operator be non-negative. Let K(X - T)-' be wmpact on P ( R x V ;dxdp) ( 1 < p < oo). Then, for any wmpact set S c R , the strong integral J, K ( x ) d x is a wmpact operator on P ( V ;dp).
Proof: We know that PK(X - T ) (v): LP (V;d p ) -+ L p ( V ;dp) is compact. Moreover, K whence
> 0 and there exists c > 0 such that 8(x,v ) 2 c on S x V
and consequently J, K ( x ) d x is compact by domination [2] [ 8 ] .0
Corollary 4.1 Let d p be a positive Radon measure with compact support V and such that the hyperplanes have zero dp-measure. We assume that the collision operator
is locally Bochner integrable and non-negative. Then K(X-T)-' is compact o n LP(R x V ; d x d p ) (1 < p < m) i f and only i f K ( x ) is compact on P ( V ;d p ) for almost all x E R.
Proof: The sufEciency part is contained in Theorem 4.1. We consider the necessity part. According to Theorem 4.3, 1
/
IB(~, B(=,r)
K ( y ) d y is compact on LP(V; dp)
for every ball B ( x , r ) c R centered at x with radius r. Letting r -+ 0 shows that K ( x ) is compact on P ( V ;d p ) at the Lebesgue points x ([53]p. 134) and consequently almost everywhere. 0
Remark 4.2 The above converse results i n LP ( 1 < p < 00) are true i f we replace K(X - T)-' by ( A - T ) - ' K . It sufices to argue by duality. They are also true i n L1 for the weak compactness but are not interesting because K(X - T)-' is not weakly compact i n general [13]. We present now a compactness result in L1 spaces. To this end we need a specific geometrical property of the measure d p
/
c15lxl5cz
LC'
dp(~)
l ~ ( t ~ ) d 0t as -+
-+
0
(4.9)
for every 0 < cl < c2 < oo and c3 < oo, where IAI is the Lebesgue measure of the set A and l Ais the indicator function of A.
60
Topics in Neutron Transport Theory
Remark 4.3 The condition (4.9) is satisfied by the Lebesgue measure on Rn or on spheres (multigroup models) . Let us show this, for instance, for the surface Lebesgue measure on the unit sphere. Let
hn-l
I(A) = Let y
= two
Then dy
00
= t n - 1 dtdS(w)
=
I(A)
1
dS(w)
At
e- 1A(tw)dt (,X> 0).
so that
e-Alyl
1 Iyl
----n=T1A(y)dy ~ 0 as
IAI
~
(4.lO)
O.
Rn
It is clear that (4.lO) implies (4.9) .
Theorem 4.4 Let dp, be a positive Radon measure satisfying the condition (4.9) and let the collision operator K be regular. (i) If f! is bounded then K('x - T)-IK is weakly compact in Ll(f! x V;dxdp,}. (ii) Iff! is bounded and convex and if a(x, v) = a(v) then K(,X _T)-l K is compact in Ll(f! x V; dxdp,) . Proof: According to Proposition 4.1, it suffices to consider Kl (,X -
T)-1 K2 where Ki (i = 1,2) have separable kernels
Then where P2 :
Iv
PI :
Iv
Thus it suffices to consider M g1 f2('x - T)jL\(S1) ' We note that h = 9112 E Ll(V ; dp,) so that, by approximation, we may assume that h is continuous with compact support on Rn . Finally, by decomposing h into positive and negative parts, we may suppose that h is non-negative. We note also that
( /\' - T)-1
S(x .v)
1 o
e -(AS+
1.8 O' (x-rv.v)dr) 0
SV
)dS
;
61
Chapter 4. Spectral analysis. A unified theory
We denote by E: Ll(n) -> £1(Rn) the extension operator (by zero) to the whole space and by R :Ll(Rn) -> Ll(n) the restriction operator. One sees that where Too is the streaming operator on Rn with collision frequency ~(v) = inf
:tEn
Moreover, if
n is
a(x, v) .
convex and a(x,v) = a(v) then
Thus
M h().. - T)jL\v) ~ RMh().. - Too )-1 E and, for convex
(4.11)
n and homogeneous collision frequency, (4.12)
Hence it suffices to prove that RMh().. - Too )-1 is compact on Ll(Rn) (and to use a domination argument in the case (i)) . We note that
Let df3s be the image of the Radon measure e-!!(v)sh(v)d/-L(v) under the dilation v -> sv. Hence
It follows that
Mh().. -
Too)-I
=
(1
00
e->'Sdf3 sdS)
*
where df3 is the Laplace transform of the family of measures df3s (8 > 0). To prove that RMh().. - Too )-1 is compact it suffices to prove that df3 is a function (see, for instance, [6] p. 74), i.e.
Topics in Neutron Transport Theory
62
which amounts to proving that d/3 is absolutely continuous with respect to the Lebesgue measure in view of the Radon-Nikodym theorem ([39] p. 117). Let A c R n then / A d/3(v)
= /
=
1A(v)d/3(v) =
/00o
/~ e-Asds /
Rn
1A(v)d/3s(v)
1A(sv)e-~(v)sh(v)dJ.L(v)
e-Asds / Rn
Since h is continuous with compact support, there exists
C2
< 00 such th~t
On the other hand, since dJ.L {O} = 0, (
llvl:5.cl
dJ.L(v)
roo e-AS1A(sv)ds +
lo
( llvl:5.C2
dJ.L(v)
roo e- AS 1A(sv)ds
lC3
is arbitrarily small, uniformly in A, if Cl is small enough and C3 large enough so it suffices that
which amounts to assumption (4.9) . <) Remark 4.4 For non-negative collision operators the property (i) is still true if K is only dominated by a regular collision operator. It suffices to use a domination argument. Thus K need not be compact with respect to velocities.
The preceding theorem provides us with sufficient (and sometimes necessary) conditions under which K(A-T)-l (resp. K(A-T)-l K) is compact in LP (1 < p < 00) (resp. is compact or weakly compact in Ll). Those conditions rely on the compactness with respect to velocities of the collision operator. Actually, for the spectral theory of T + K, we only need some power of K(A - T)-l be compact. Thus, it seems natural to try to weaken the assumptions on K. It is plausible that the optimal assumption is that K be a power compact operator in LP(V;dJ.L) . This conjecture is suggested by the following converse result.
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Chapter 4. Spectral analysis. A unified theory
Theorem 4.5 Let the collision operator be non-negative and independent of the space variable. Let [(A- T ) - ~ K ] be compact in P ( R x V ;dxdp) ( 1 < p < oo) (resp. weakly compact in L 1 ( R x V ;dxdp)). Then K n is compact in P ( V ;d p ) (resp. weakly compact i n L 1 ( V ;d p ) ) . The proof is based on the following technical result. For each S C R we set
Then
Lemma 4.1 Let V be bounded and let S1, S2 be two open subsets of R such c S 2 , C R. Then there exists a constant c = c(S1,S2) such that that
Proof: Let $ E LP(R x V ;dxdp), $
Jsz Jo
> 0 and cp = Q
s(x'v)
=
dx
e
s l ~Then
-(A8+Ji s(z-rv,v)dr) cp(x - sv, v)ds
and the last term is bigger than
Finally Ps2 ( A - T ) - l Q s , $ is bigger than
where vo is the highest speed. Now, if xi E S1 and lsvl dist Sl;aSz) ) then x' E S2 - sv. Let particular if s < ( vo
< dist(Sl; as2)(in
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Topics in Neutron Transport Tbeory
The last term in (4.13) is larger than
{d e-(Mllulloo)sds { 'ljJ(x', v)dx'
lo
lSl
= cPs1'ljJ
which ends the proof. <> Proof of Theorem 4.5 : Let SI C S2 C .. . C Sn that Si c SiH (1 :S i :S n - 1) and
=
S
eSc n be such
J= [('\-T)-IKr ·
Let
J be the restriction to V(V)
V(V) (1 < J
p
of PSn J. We note that J is compact on
< 00) (resp. weakly compact in L1(V»
= PSn (.\ - T)-1 K ... (.\ - T)-1 K ~ PSn (.\ - T)-IQsn_l K(.\ - T)-IQsn_2K . . . (.\ - T)-IQ Sl K
because Qs; :S I. According to Lemma 4.1, PSn (.\ - T)-IQsn_l ~ cnPSn_ 1
and PSn- 1K
= K PSn - 1.
(c n > 0)
It follows that
By using Lemma 4.1 repeatedly we get
J ~ Cn Cn-l ... C2 Kn Ps 1. On the other hand, the restriction to V(V) of PS 1 is nothing but whence n
J ~ cKn
i
c=
ISll I
ISll II Ci· i=2
n
Finally K is compact on V(V) (1 < p < 00) (resp. weakly compact in L1(V» by domination. <> We end this section by showing however that, under additional conditions, the powers of (.\ - T) -1 K are not smoother than (.\ - T) -1 K (as regard to compactness). Theorem 4.6 We assume that the measure dJ.L is invariant by the transformation v -+ -v and that p = 2. Let K be a positive self-adjoint collision operator mapping L2(n x Vi dxdJ.L) into the subspace of even functions (with respect to velocities) and that a(x,v) = a(v) = a(-v). Let some power of ('\_T)-l K be compact inL2(n x Vi dxdJ.L) then ('\_T)-1 K is also compact.
65
Chapter 4. Spectral analysis. A unified theory
a
Proof: Let be the positive square root of K . By assumption @(AT ) - ' a is power compact. We are going to show that *(A - T)-'* is self-adjoint for real A. Let f , g E L ~ ( Rx V ;dxdp). Then (@(A T)-'f , g ) is equal to
a
We note that the ranges of operators K and have the same closure because K and have the same kernel. Hence a g ( s .) , is even. In view of the evenness property of a ( v ) ,the last integral is equal to
I
e - ( X + u ( v ) ) S f i f( x + sv, v)ds &(x,
v),
i.e.
( ( A - T * ) - l f i f ,&g)
=(
f , &(A
-T
)-'G).
T)-'a
Being self-adjoint and power compact, & ? ( A is in fact compact. Actually (A-T)-'* is compact. To see this, let { fm) c L 2 ( R x V ;dxdp) be a sequence converging weakly to zero and let gm = ( A - T)-l&fm. We note that
where A* is the infimum of a(.).By replacing g by gm in (4.14) we get
(A
+ A*) llgm1l2 < I (*fin,( A -
~)-'*fm)
1
Hence gm -+ 0 strongly because *(A - T)-'@fm -, 0 strongly and this shows the compactness of (A - T)-'@ and ( A - T ) - ' K . 0
4.3
Evolution problems in LP ( l < ~ < c a )
Let K be a collision operator and { V ( t ) ;t 2 0 ) be the co-semigroup generated by T K . This section is devoted to the spectral analysis of V ( t ) . More precisely, we are interested in giving sufficient conditions in terms of ( d p ,K ) under which the essential types of { U ( t ) ;t 0 ) and { V ( t ) ;t 2 0 ) coincide. This will imply that outside the disc {P; [PI 5 evt) (where 7 is
+
>
66
Topics in Neutron Tkansport Theory
the essential type of the streaming semigroup {U(t);t 2 0)) the spectrum of V(t) consists of (at most) isolated eigenvalues of finite algebraic multiplicities providing us with an asymptotic expansion of V(t) as t -+ oo. The functional analytic theory is given in Chapter 2. Our main object is to apply it to concrete transport semigroups. We recall that V(t) is given by a series m
where
On the other hand, according to the theory in Chapter 2 (Lemma 2.1), the m-th remainder term of the Dyson-Phillips expansion is given by
with the notation [f] = f , [f12 = f * f = f ( t - S) f(s)ds , [flm = f * f *. . .* f (m times) where f : [0,oo[ + L(X) is strongly continuous. We recall, according to Theorem 2.10, that the stability of the essential type is ensured if some remainder term &(t) is compact in X = LP(R x V; dxdp) (1 < p < oo) or weakly compact in L1(R x V; dxdp). Thus our main concern here is to study the compactness of the remainder terms L ( t ) (m 2 1). The present section is devoted to the case 1 < p < oo. We begin with
Theorem 4.7 Let 1 < p < oo and let the collision operator K be regular. We assume that the measure dp is such that, for every bounded set D c Rn,
b
e-ix.Wdp(x)-+ 0 as
IwI
-+
oo.
(4.15)
(i) If R is bounded then R2(t) is compact in LP(R x V; dxdp) (iz) If R is convex (not necessarily bounded) and if u(x, v) = u(v) then R2(t) is continuous in t 0 for the uniform topology.
>
Proof: We consider first the case p = 2. We recall that R2(t) is compact ~ is compact (resp. continuous in t > 0 for the uniform topology) if [ u K ](t) (resp. continuous in t > 0 for the uniform topology) in view of Theorem 2.6 (resp. Theorem 2.7). On the other hand, [ u K ] ~ = (UK) * (UK) = U* (KUK) whence the same arguments enable us to consider only KU(t)K
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Chapter 4. Spectral analysis. A unified theory
(t > 0). Moreover, since the collision operator is regular, we may argue by approximation (Proposition 4.1) and restrict ourselves to K 1 U(t)K2 where Ki (i = 1,2) have kernels of the form
We note that K 1 U(t)K2 is factorizable as follows
where 02 : 'P E L2(f2 x V) ----> Q2(X)
i
'P(x, v')g2(v')dJ.L(v') E L2(f2),
S(t) is defined by 'Ij; E L2(f2) ---->
i
h(v)e- J:u(x-7"v,V)d7"'Ij;(x_tv)x(t < s(x,v))dJ.L(v) E L2(f2)
(where h(v) = gl(v)h(v)) and 0 1 : 'Ij; E L2(f2)
----> Ql (.)JI (.)'Ij;(.) E
L2(f2 x V).
Hence it suffices to prove that Set) (t > 0) is compact (resp. Set) depends continuously on t > 0 in the uniform topology). By approximating h E Ll(V; dJ.L) by continuous functions with compact supports we may suppose that h is continuous with compact support. Finally, by decomposing h into positive and negative parts, we may suppose that h is non-negative. A basic observation is Set) ~ RSoo(t)E where E: L2(f2)
---->
L2(Rn)
is the extension operator (by zero),
is the restriction operator and
where
Q(V)
= inf a(x,v). xEO
68
Topics in Neutron Transport Theory
Moreover, if n is convex and if a(x , v) = a(v) then (4.16)
Set) = RSoo(t)E.
Thus we are led to show the compactness of RSoo(t) when n is bounded and to appeal to a domination argument [2J . We will also show that Soo(t) is continuous in t > 0 for the uniform topology and this will prove the second claim in view of (4.16). Another basic observation is that, for 'l/J E L2(Jr'),
Soo(t)'l/J
J J
h(v)e-t!l.(v)'l/J(x - tV)d/-L(v)
=
'l/J(x - z)dVt(z) = 'l/J * dVt
=
(4.17)
where dVt is the image of the Radon measure h(v)e-t!l.(v)d/-L(v) under the dilation
v
--+
tv.
According to Lemma 3.2, the compactness of RSoo(t) relies on the condition II
Th(SOO(t)'l/J) - Soo(t)'l/JII
L 2(Rn)
--+
0 as h
--+
for any bounded B c L2(Rn), where Th'l/J(X) Fourier Transform, this amounts to (e ih .( -1)S;::W'l/J11 II
--+
0 as h
--+
0 uniformly in'l/J E B
= 'l/J(x + h). By using the
0 uniformly in 'l/J E B .
L 2(R,)
It is easy to see that there exists a constant c
> 0 depending on B such
that (e ih .( -1)s::;t)'l/J112 ::; 4 II
r
le ih .( - 11
V A> O.
uniformly in 'l/J E B .
(4.18)
IS;::W'l/J12 d(+c sup
J1(I>A
Thus it suffices to prove 2 Is::;t)'l/J1 d(
r
--+
J1(I >A
0 as A
I(I
--+ 00
We note, according to (4.17) , that
S;::W'l/J
= (27r)~ ;j;«()d,;,.«() = ;j;«() =
;j;«()
J
J
C iz'(dVt(z)
citv·(h(v)e-t!l.(v)d/-L(v).
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Chapter 4. Spectral analysis. A unified theory
Let C be the support of h. By assumption the Fourier Transform of lcdf,L goes to zero at infinity. Since h(v)e-t~(v)df,L(v) is absolutely continuous with respect to lcdf,L, then (see [14] Proposition 1, p. 31) the Fourier Transform of d{3(v) = h(v)e-t~(v)df,L(v) goes also to zero at infinity. Hence
which proves (4.18) and the first claim. We consider now the continuity (in t > 0) for the uniform topology of
Let t > 0 be fixed. Then
IISoo(t)1/! -
II1/! * (dVt -
Soo(t)1/!IIL2(Rn)
< whence
IISoo(t) -
Soo(t) II
sup (ERn
dl't)II£2(Rn)
Idv't() - dl't() I II 1/! II £2 (Rn)
::; (~UXn IdVt() - iz;-(Ol·
On the other hand,
splits as
It is clear that h () -+ 0 as t -+ t uniformly in ( E Rn. Moreover , h(() and h() are arbitrarily small for 1(1 large enough, uniformly in t belonging to a fixed small neighborhood oft, because the Fourier Transform of h(v)e-t~(v)df,LC1!) goes to zero at infinity. Finally, it is clear that h(()h (() -+ 0 as t -+ t uniformly in ( bounded. HeI1ce
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Topics in Neutron Transport Theory
which ends the proof of the theorem for p = 2. The case 1 < p < 00 is tackled as follows. We first note that R 2 (t) E L(£P(n x V)) depends continuously on K E L( £P (0. x V)) (uniformly in bounded t). Therefore it suffices to give a proof for a smooth collision operator K, say with a kernel of the form iEI
where I is finite, (ti E LOO(n) and Ii, 9i continuous with compact supports. In such a case R2(t) E L(£T(n x V)) for alII :-:; r < 00 and consequently the above L2 results extend to £P spaces (1 < p < 00) by interpolation. 0
Remark 4.5 We point out that (4.15) is satisfied for the Lebesgue measure on R n (Riemann-Lebesgue Lemma) or on spheres (see, for instance, W. Littman [25]) . The convexity of 0. is essential for the time continuity of R2(')' It is possible however to remove this assumption by imposing a stronger hypothesis on the measure dJ.L (see Corollary 4.3) . Before proceeding further, we point out an interesting by-product of the previous proof. It is a measure theory result which is not (to our knowledge) in the current literature although it looks quite reasonable. In one dimension it is contained in the well-known Wiener's theorem ([20J p. 138) (see also [41J p. 32) . It could probably be deduced from the onedimensional result. It is surprising to prove it so indirectly (via Transport theory!)
Corollary 4.2 Let dJ.L be a bounded Radon measure on Rn whose Fourier transform vanishes at infinity. Then the hyperplanes have zero dJ.L-measure. Proof: Let 0. be bounded and convex. We showed, in the proof of the previous theorem, that K I U(t)K2 is compact in L2(n x V) for t > 0 (for any regular collision operators KI and K2)' It follows that
1
00
K I ()..-T)-IK2 =
e->.tKI U(t)K2dt is compact in L2(n x V).
By choosing K2 = (KI)*' the arguments in the proof of Theorem 4.6 show that K I ().. - T)-I is compact in L2(n x V). This implies, according to Remark 3.1 and Remark 4.1, that the hyperplanes have zero dJ.L-measure which ends the proof. 0 We give now an extension of Theorem 4.7
Theorem 4.8 Let 1 < p < 00 and let the collision operator be regular. We assume that the measure dJ.L is such that translated hyperplanes have zero dJ.L-measure and that 0. is bounded. Then the remainder term R3(t) is compact.
Chapter 4. Spectral analysis. A unified theory
71
Proof: We only consider the case p = 2; the general case follows by approximation and interpolation as in the proof of Theorem 4.7. We deal now with R3(t) = [UK]3 * V. It suffices to prove the compactness of [UK]3. Actually, since [UK]3 = [UK] * [UK]2 = U * (K [U K]2), it suffices to show that
K[UK]2= lotKU(S)KU(t-S)KdS is compact. By approximating the regular collision operator K, it suffices to consider
lot K 1 U(s)K2U(t - S)K3ds where Ki (i = 1, 2,3) have kernels
In such a case,
where 02 E
L(L2(n x V), L2(n))
S2(t) E L(L2(n), L2(n)) Sl(t) E L(L2(n), L2(n))
01
E
L(L2(n), L2(n x V))
act , respectively, as follows
cp
-t
f
[
D:2(X) V h2(V) e
f
[
cp-t . vh1(v) e
cp
-t
_It
D:l(x)cp(x)h(v)
0
_It
U(X-Tv,v)dT
0
U(X-Tv,v)dT
1
cp(x - tv)x(t < s(x, v))
1
cp(x-tv)x(t < s(x,v)) dp,(v)
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Topics in Neutron Transport Theory
and h2 = g2/J, hI = glh By approximating hi (i = 1,2) by continuous functions with compact supports (and decomposing them into positive and negative parts) we may suppose that hi (i = 1,2) are continuous functions with compact supports and non-negative. Thus we are led to prove the compactness of lot SI(s)S2(t - s)ds.
It is not difficult to see that
where
c = IIa31100 IIhI li oo IIh21100 and e' SUpphi C
is a constant such that
{v; Ivl::; e'}
(i = 1,2) .
We denote by Rand E, respectively, the restriction operator to n and the extension operator to Rn. Thus, by domination [2] , it suffices to prove the compactness of Riot S(s)S(t - s)ds.
We introduce the measure df3(v) S(t)cp
=
J
(4.19)
= 1{l v l:5 c ,}d/-l(v).
cp(x - tV)df3(v)
=
Then
J
cp(x - z)df3t(z)
where df3t is the image of df3 under the dilation v (4.19) is nothing but
--t
= df3t * cp
tv. Thus the operator
where da
= lot df3s * df3t-sds.
(4.20)
Arguing as in the proof of Theorem 4.7, it suffices to show that
(4.21)
73
Chapter 4. Spectral analysis. A unified theory where
z is the Fourier Transform of the measure da. We note that
z ( < )=
bf
[I
[J
~ ( ( ) d ~ ~ (=~ o) d se - h . c d ~ s ( ~ ) ] e-'~.~d/3~-,(y)]ds,
Thus
z(O
=
/J
e-fi.C - e-"z.C i("
-
y1.C
Introducing the polar coordinates C = ICI w , w as
dP(x)dP(~). E
(4.22)
Sn-l, we decompose (4.22)
Hence
where
Thus, to prove (4.21) , it suflices that dP(x)dp(y) = 0 uniformly in w E Sn-'. We note that
so, by the dominated convergence theorem, it suffices to show that for each y E Rn SUP
wESn-'
1
l(z-y).wll€
dp(x) -,o as E
-,0.
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Topics in Neutron k s p o r t Theory
Using the compactness of 9 - l ( a s in Lemma 3.1), this amounts to
J
dp(x) -+ 0 as E -+ 0 for each w E sn-land y E Rn
I(x-Y).~~E
and this means that dp {x; (x - y).w = 0) = 0,i.e. translated hyperplanes have zero dp-measure. Findy, this is equivalent to saying that translated hyperplanes have zero dp-measure. 0 The previous theorems provide us with sufficient conditions for the compactness of R2(t) or R3(t). Actually, it is possible to give a general condition under which &(t) (m 2) is compact. However, this condition is of abstract character. This is the reason why we state this general result separately. To this end we introduce some notations. For each c > 0 we d e h e a truncation of the measure dp
>
Let M(Rn) be the space of bounded Radon measures on Rn. For each couple t ~ ( 0 , o o ) - - + d a i ( t ) € M ( R n )i = 1 , 2 of measure-valued and strongly continuous mappings, we define their convolution as t dal * do? = dal(s) * da2(t - s)ds
1
where the sign * under the integral sign is the usual convolution of measures. We can also define the repeated convolution of t E (0, oo) + da(t) E M(Rn) and set
[da(t)lm= da * . . - * d a (m times).
Then we have
.- Theorem 4.9 Let 1 < p < oo and let the collision operator be regular. We assume that R is bounded. For each 0 < c < oo we define dp(v) by (4.23). Let dBt be the image of dp under the dilation v -+ tv (t > 0). If there exists an integer m 2 2 such that the Fourier Transform of [dptlm-I vanishes at infinity, then &(t) is compact.
Proof: As previously, we only consider the case p = 2. Since &(t) [UKIm* V , it suffices to show the compactness of
=
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Chapter 4. Spectral analysis. A unified theory
Again, it suffices to prove the compactness of K [U K]m-l . On the other hand, by the usual procedure, it suffices to consider (4.24) where ki(x,v,v') = ai(x)fi(v)gi(v')
(1 ~ i ~ m)
with ai E Loo(n); Ii,gi continuous with compact supports and nonnegative. Let c > 0 be such that the supports of Ii, gi (1 ~ i ~ m) are included in {v; Ivl ~ c } and let m-l
'Y
Then
=
m-l
m-l
(II lIaill II Ilhll II Ilgill i=2 i=2 i=2 oo )(
Kl [U K 2] * ...
oo )(
* [U Km]
oo )'
~ 01R [d,Bt]m-l E02
where [d,Bt]m-l denotes symbolically the operator of convolution with the measure [d,Bt]m-l, O 2 : 'ljJ E L2(n x V)
-+
am (x)
J
'ljJ(x, V')gm(v')dp,(v') E L2(n)
and 0 1 : t.p E L2(n)
-+
al(x)fI(v)t.p(x) E L2(n x V).
Finally it suffices to prove that R [d,Btl m -
1
:
L2(Rn)
-+
L2(Rn)
is compact
and this is a consequence of the assumption on the measure [d,Bt]m-l . <> We close this section with a converse theorem in the spirit of Theorem 4.5. We note that if, for some integer m, Rm(t) is compact in V(n x V; dxdp,(v)) for all t > 0 then (see Theorem 2.8) [(>. - T)-1 K]m+1 is compact in V(n x V; dxdp,(v)) and consequently, by Theorem 4.5, Km+l is compact in V(V; dp,) . Actually the following stronger result holds Theorem 4.10 Let 1 < p < 00 and let K E L(V(V)) be non-negative. We assume that V is bounded and that 0. is convex and bounded. Let there exist to > 0 and m ~ 1 such that Rm(t) is compact in V(n x V;dxdp,(v)) for t E [0, to] . Then K m is compact in V(V; dp,).
Before giving a proof, we need a preliminary result. For each Sen, we define the multiplication operator Qs : 'ljJ(
)
-+
ls(x)'ljJ(x, v) .
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Topics in Neutron nansport Theory
Lemma 4.2 Let S1 and S2 be two open subsets of R such that Then, for all $ E LP+( V ;d p ) ,
QslU(s)Qs2$ > e-sllullm &sl$ i f s l
as2)
dist ( S 1 ;
vo
C
Sz.
(4.25)
where vo is the maximum speed. Proof: Let $ E LP+(V;d p ) and let cp = Qs,$J. Then
It is clear that if x t S1 and if s 5
d i s t ( ~ a s 2 )then
x - sv E S2. Hence
which ends the proof. 0 Proof of Theorem 4.10 : Let {So,...,Sm+l) be m+2 convex open subsets of R such that for (i = 0, ...,m). Let
c SSil
Let T 5 min(to, c). By assumption, &@) is compact in L P ( R x V ;d x d p ( v ) ) . Moreover
k@) = [UKIm*V > [UKIm*U
because Qsi 5 I and V 2 U . According to Lemma 4.2,
QS,U(S)QS,+~~L~(1 V )e-sl'ullm Qs, . Hence
77
Chapter 4. Spectral analysis. A unified theory Similarly, the last term of (4.26) dominates
Qs",_, [e- sllulI "",
r
K.
By induction, one sees that
Rm(I)ILP(V) ~ Qso [e-sllull"",]m (I)Km
* ... * (e-sllull"",)
where [e- sllull "",] m = (e-sllull"", )
PSo : 'Ij; E £P(n x V)
---->
(m times) . Let
r 'Ij;(x, v)dx
iso
E £p(V) .
Then the compact operator PsoRm(I)ILP(V) dominates the following operator meas(So) [e- sllulI "", m (I)Km
1
and consequently K m is compact in £p(V; df-L) by domination [2] . (;
4.4
Evolution problems in L1
We begin with a continuity result which does not rely on the convexity of
n. Proposition 4.2 Let K be a regular collision operator on Ll(n x V). Let
a(x, v) For each 0
= a(v)
and t
-t
a(tv) be continuous for each v -j. O.
< c < 00 we set d{3(v) t
= l{lvl~c}df-L(v) and assume that
> 0 - t d{3t
is continuous in the uniform topology (i. e. the norm of total variation) where d{3t is the image of d{3 under the dilation v - t tv. Then R 2 (t) is continuous in t ~ 0 for the uniform topology.
Before proving this result, let us show that the Radon measures satisfying the assumptions above "are" those which are absolutely continuous in "speeds" but arbitrary in "angles". This excludes the usual multigroup models. Proposition 4.3 Let df-L(v) = da(p) ® d{3(w) where v = pw, w E sn-l, d{3 is a Radon measure on Sn-l and da(p) = h(p)dp (h E £1(0, (0)). Then t > 0 - t dp,t is continuous in the total variation norm.
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Topics in Neutron Transport Theory
Proof We note that IId/ltll :::; Iid/lil :::; IIhll£l(O.oo) Ild,Bll. Thus d/lt depends (linearly and) continuously on h E Ll(O,oo), uniformly on t > O. Then, by approximation, we may suppose that h is continuous with compact support. We fix t > 0 and take continuous test functions cp with compact supports, (d/lt,cp)
=
J
cp(v)d/lt(v)
=
J
(d/l t , cp)
=
IsSn-l
Thus
and
(d/lt - d/lr , cp) =
1 hn-l 00
cp(tv)d/l(v)
=
1
00
d,B(w)
JJ
cp( TW)
0
cp(tpw)h(p)dpd,B(w).
T dT cp(Tw)h( - ) t t
[rl h(~) - rl h(1)] dTd,B(w) .
Therefore
Iid/lt-d/lrll:::; roolrlh(!.)_rlh(~)ldT.
Jo
t
t
r
Jsn-l
d,B(w)---->O ast---->t
in view of the smoothness of h. <> Remark 4.6 The assumption in Proposition 4.2 is not satisfied by the Lebesgue measure on spheres. Indeed, let (for instance) d/l be the Lebesgue measure on sn-I and let 0 < t < t. We choose a continuous function cp with compact support such that Icp(z)1 :::; 1, cp(z) = 1 if Izl = t and cp(z) = -1 if Izl = t. Then
and d/lr(cp)
so
that Iid/l t
-
= hn-l cp("tw)d/l(w) = -Isn-Il
d/lrll = 2lsn-11 Vt < t
Proof of Proposition 4.2 : We know that R 2(t) = [U K]2 thanks to Lemma 2.3, to prove that t > 0 ----> KU(t)K
* V. It suffices,
is continuous in the uniform topology.
By approximating K by separable kernels, we are led to study K I U{t)K2 where ki(x, v, Vi) = O!i{X)fi(V)9i{V') (i = 1,2), O!i E LOO{n), Ii E LI{V), gi E Loo(V) . We factorize K I U(t)K2 as 0IS(t)02 where
02: 'I/J E LI(n x V) ----> 0!2{X)
Iv
'I/J(x, V' )g2(V')d/l{v') E LI{n),
79
Chapter 4. Spectral analysis. A unified theory Set) is the operator cp E L1(n)
--+
Iv
e-tCT{V')cp(x - tv')h(v')l{t<s{x.v,)}dl.£(v') E L1(n)
(h = g112) and 0 1 : cp E L1(n)
--+ Ct1 (x)cp(x)h (v) E
L1(n x V) .
Finally, it suffices to study Set) and to assume that h is continuous with compact support. Let t > 0 be fixed. We note that
S(t)cp
J =J =
e-tCT{v')cp(x - tv')h(v')l{t<s{x,v,)}dl.£(v') e-tCT{v')cp(x - tv')h(v')1{1<s{x.tv,)}dl.£(v')
so that
S(t)cp
J =J =
cp(x - z)1{1<s{x.z)}e- tCT {f) h( I )dl.£t(z)
cp(x - z)1{1<s{x.z)}q(t, z)dl.£t(z)
where dl.£t is the image of dl.£ under the dilation z e-tCT{f} h( I) ' Hence
II S(t) - S(1)11 L{L'{!1) ::; Ilq(t, z)dl.£t(z) -
--+
tz and q(t, z)
q(t, z)dI.£I(z)
II
(4.27)
where the right-hand side of (4.27) (a total variation norm) is estimated by
Ilqllv'" IIdl.£t - dl.£III
+
J
Iq(t, z) - q(t, z)1 dl.£t(z)
(4.28)
where the first term of (4.28) goes to zero as t --+ t by assumption while the second one goes to zero by the dominated convergence theorem. <:; We are in position to complement the continuity result of Theorem 4.7 with Corollary 4.3 Let 1 < p < 00 and let K be a regular collision operator on V(n x V) . We assume that dl.£ satisfies the condition of Proposition 4.2. Then t ~ 0 --+ R2(t) E L(V(n x V))
is continuous for the uniform topology.
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Topics in Neutron Transport Theory
Proof: Since R2(t) depends continuously on K and uniformly in bounded t, it suffices (by approximation) to assume that K is smooth. In such a case R2(t) maps £T(O x V) into itself for any 1 ::; r < 00 and the result follows from Proposition 4.2 by interpolation. <:) We turn now to compactness problems in Ll. We begin with
Theorem 4.11 Let K be a regular collision operator in Ll(O x V) and let d/L(v) = dv . We assume that 0 is bounded. Then R2(t) is weakly compact in Ll(O x V). In addition, if 0 is convex and if c>(x, v) = c>(v) then R2(t) is compact in £1(0 x V).
Proof Since R 2(t) = [U K]2 * V and [U K]2 = U * (KU K) it suffices, thanks to Theorem 2.2 and Theorem 2.3, to consider KU(t)K (t > 0). By the usual approximation procedure, we may restrict ourselves to K 1 U(t)K2 where
By factorizing K 1 U(t)K2, as in the proof of Proposition 4.2, it suffices to consider the operator Set)
where (h = gl12). By decomposing h into positive and negative parts, we may assume that h is non-negative. Let
S(t)
r
iv
(4.29)
We note that the last integral operator in (4.29) is compact in £1(0) because rnh(t) is integrable (see, for instance [6], p. 74) whence Set) is (at least) weakly compact by domination. If 0 is convex and c>(x, v) = c>(v), then
S(t)
Iv
h(v)e-tu(v) E
where E is the trivial extension operator to R n and R is the restriction operator to O. Hence it suffices to prove the compactness of
81
Chapter 4. Spectral analysis. A unified theory l.e.
->
R
r
JRn
Cnh(x - x' t
)e-t
This follows from the fact that t-nh(t)e-t
0
Remark 4.7 The same arguments show that, in V (1 < p < 00) , R2(t) is compact when dp,(v) = dv . In the previous theorem, the weak compactness of R2(t) is deduced from that of KU(t)K (t > 0). This result is probably not true for the usual multigroup models as is suggested by the following result.
Proposition 4.4 Let dp, be the surface Lebesgue measure on the unit sphere sn-l . Let n be convex and let d be its diameter. Then S(t) = 0 if and only ift > d and S(t) is not weakly compact in U(n) when t < d. Proof: We assume, without loss of generality, that (dp,) a .e. Then S(t)
kn-l
(J
= O. Let h
>0
tv')h(v')I{t<s(x,v,)}dp,(v') .
It is clear that s(x, v) < d and then S(t) = 0 if t > d. Assume that t < d and let x, x" E be such that t < Ix - x*l. Then s(x, v) > t for v = I~=~:I whence the set {(x,v) x sn-I jS(X,v) > t} is not empty and is open because s(. ,. ) is lower sernicontinuous (see [51]). Thus, there exists Den and We sn-I, open sets, such that s(x, v) > t on D x W . Let
n
En
S(t)
r
kn-l
h(v)
~
r
Jw
h(v)dp,(v)
>0
which shows the first claim. We are going to show that S(t) is not weakly compact (t < d) . We assume, for the sake of simplicity, that h = 1. Thus
Let
~
0 and A
c
n be a measurable subset, then
r dp,(v) JAr
tv)l{t<s(x ,v)}dx .
We note that
r
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Topics in Neutron Transport Theory
so that
rS(t)cpdx inrcp(z)dz isn-ln(A;Z) r l(t<s(z+tv,v»d/-L(v) =
iA
and consequently (4.30) According to the Dunford-Pettis criterion of weak compactness [10], it suffices to show that the right-hand side of (4.30) does not go to zero as the Lebesgue measure of A goes to zero. To this end we proceed as follows: Let x, x' E 0 be such that x' = t. We may suppose, without loss of generality, that x = O. We define
Ix - I
.......
D
J
= {x
x , ,-- } Ix I= t} , D = {t> xED . J
E 0;
::::::.
We note that jj C sn-l and that s(tv, v) perturbation argument shows that
> t \:Iv
E
D . Actually, a simple
s(z + tv, v) > t \:Iv E jj and z small enough. Let
Am
=
{
X
,
,-
E O;dist(x ,D)
<
1 }
m
.
Clearly Am is an open neighborhood of D whose Lebesgue measure _goes to zero as m -+ 00 . It follows that A'1- z is an open neighborhood of jj for z small enough (the smallness of z depending possibly on m). Thus, for m fixed,
in a neighborhood of z
= O.
Finally
despite the fact that the Lebesgue of Am goes to zero. 0 We end this section by an extension of Theorem 4.11 covering the multigroup models
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Chapter 4. Spectral analysis. A unified theory
Theorem 4.12 Let D be bounded. We assume that K is a regular collision operator in Ll(D x V) or is non-negative and dominated by a regular collision operator. Let dJl be such that for every 0 < Cl < C2 < 00 and t > 0
where IAI is the Lebesgue measure of the set A . Then R3(t) is weakly compact in Ll(D x V) . Proof Let K be a regular collision operator. We observe that R3(t) = [UKj3 * V and [UKj3 = U * (K [UKj2) so, according to Theorem 2.3 (or Theorem 2.4), it suffices to prove the weak compactness of
lot KU(s)KU(t - s)Kds . By approximating K by separable kernels, we may restrict ourselves to
lot K U(s)K2U(t - s)K3ds 1
(4.32)
where ki(x, v, v') = Cti(X)Ii(V)gi(V'), Cti E LOO(D), Ii E Ll(V), gi E Loo(V) (i = 1,2, 3) . One sees that (4.32) is factorizable as
where
and Si(S) E L(Ll(D)) (i = 2,1) act, respectively, as
r.p
-+ Ct2
r.p while
-+
( )ivrr.p( )-is -i s ivr X
x - sv e
r.p(x - sv)e
a
a
u(x-rv,v)dr
u(x-rv,v)dr
( ) 1(s<s(x,v»h 2 v dJl(v)
l(s<s(x,v»hl(V)dJl(v)
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Topics in Neutron Transport Theory
(hI = gIi2 ,h2 = g2!3 E LI(V)) . It suffices to consider lot SI(S)S2(t - s)ds.
(4.33)
By approximation (and decomposition) we may assume that hi (i = 1,2) are continuous with compact supports and non-negative. One verifies that the operator (4.33) is dominated by the operator
r.p E LI(fl)
--+
R (lot d/it * d/it-sdS)
* Er.p
where Rand E are the usual restriction (to fl) and extension (to Rn) operators,
d/i = l{lvl:5c}dp,(v), the constant c being such that supp(h i ) C {Ivl ::; c} (i = 1,2), * denotes the convolution of measures and d/it is the image of d/i under the dilation v --+ tv. It suffices to show the compactness of
r.p E LI(Rn)
--+
R (lot d/it * d/it- sdS)
* r.p E LI (Rn) .
To this end it suffices that J~ d/it * d/it-sds be a (U) function (see, for instance, [6] p. 74). By the Radon-Nikodym theorem ([39] ,p. 117), this amounts to
i.e.
which means that
i.e.
r i1xl.lyl :5C
[ rt 1A(sx
io
+ (t -
S)Y)dS] dp,(x)dp,(y)
--+
0 as
IAI
--+
O.
(4.34)
By the fact that dp, {O} = 0, one sees that (4.34) is equivalent to the assumption (4.31). This ends the proof when K is regular. If K is nonnegative and (only) dominated by a regular collision operator, the result follows from the previous one by a domination argument. 0
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Chapter 4. Spectral analysis. A unified theory
4.5
The effects of delayed neutrons
The transport equations considered in the previous sections describe scattering phenomena and also the production of additional neutrons by fissions provided they appear instantaneously (prompt neutrons). However , in fissile materials some neutrons may appear after a time delay as the decay products of radioactive fission fragments. In such a case, the appropriate equation of the neutronic distribution is the following 8fo 8fo at + v . 8x + a(x, v)fo
J
k o(x , v,v' )fo(x , v' , t)dJ1-(v')
+ fAdi
(4.35)
i =l
subject to boundary and initial conditions fo( .,. , t)IL = 0, fo(x, v , 0) = fO(x, v)
where the distributions fi (1 ~ i ~ m) of the delayed neutron emitters (precursors) are governed by the differential equations dfi dt fi(x , v , 0)
-Adi
+
J
ki (x , v , v')fo(x,v',t)dJ1-(v')
fi(x , v) ; (1 ~ i ~ m)
(4.36)
and the positive numbers Ai are the radioactive decay constants [26J . It is possible to solve explicitly fi (1 ~ i ~ m) in terms of fo by means of (4.36) and to insert their expressions in (4.35) . This enables us to deal with only one equation. However, such an equation containing a memory term is not well-suited to the spectral theory. Thus it is more convenient to deal with the coupled system (4.35) (4.36) . To show its well-posedness we write it in the vector form
dw
dt = Aw
; w(O)
= Wo
(4.37)
where W = (Jo , iI , ... , fm)J. and A is the (m+ I, m+ I)-matrix of operators
A=
Topics in Neutron Transport Theory
86
:s :s
where To is the streaming operator defined in section 4.1, Kj (0 j m) represent the integral operators with kernels k j (x, v, v') and I is the identity operator. The functional setting is [£P(r! x V)Jm+l ; 1
:s p <
00.
It is convenient to split the matrix A as
where
To 0 T=
o
(
-All 0
;B=
000
We note that T, with domain
D(T) = D(To) x [£P(r! x V)Jm , generates the explicit co-semigroup
.T. = 'J!
(
f~m~
Uo(t)fo ) )
E
where
Uo(t)fo = e -
[£P(r! x V)Jm+l
-t
e-A1th
U(t)\lf = (
J:
a(x-rv,v)dr fo(x
e-A",tfm
- tv, v)X(t < s(x, v)).
:s :s
Finally, if K j (0 j m) are bounded in V(r! x V) then the Cauchy problem (4.37) is well-posed and governed by a CO-semigroup {V(t); t ~ O}. The main purpose of this section is the spectral analysis of A and {V(t); t ~ O} by exploiting the compactness results of the previous sections. For the simplicity of presentation, we restrict ourselves to the case p > 1. We begin with the spectral properties of the generator A . We assume that r! is bounded and denote by ry the (essential) type of {Uo(t); t ~ O} . We are interested in
a(A) n {>";ReA > ry}. We are going to show how the decay constants Ai affect the spectrum of A. To this end we define the sets E = {-Ai; -Ai> ry} ; M = {A; ReA> ry, A fj. E}
and assume that E is not empty in order to fit the physical situations.
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Chapter 4. Spectral analysis. A unified theory
Theorem 4.13 We assume that the measure dp, is such that the hyperplanes have zero dp,-measure and that the operators K j (0 :s; j :s; m) are regular in the sense of Definition 4.1. Then a(A) n M consists oj, at most, isolated eigenvalues with finite algebraic multiplicities.
Proof We consider the problem Aw-Aw=
>.fo - [Tofo
+ Kofo + L
Adi]
i=l
Ali - [-Adi
+ Kdo]
=
gi ; (1 :s; i :s; m)
(4.38)
It is easy to see that (4.38) is equivalent to
fo - (A - TO)-l K(A)fo
Ii =
gi +Kdo A + Ai
(1 :s; i :s; m) .
(4.39)
According to Theorem 4.1, (A - TO)-l K(A) is compact in lJ'(D. x V) (1 < p < 00). By Gohberg-Shmulyan's theorem, [I - (A - TO)-l K(A)rlexists except (possibly) for a discrete set consisting of degenerate poles. This implies that (A - A)-l exists for A E M except (possibly) for a discrete set of eigenvalues with finite algebraic multiplicities. <> We point out that, in general, the constants -Ai (1 :s; i :s; m) belong to a(A) . Indeed Theorem 4.14 Let j E [1 , .. ., m] and A = -Aj . (i) If>. E ap(A) then 0 E ap(Kj). Conversely, if Ker(Kj)nD(To) -=f. {O} then A E ap(A) . (ii) If 0 E ar(Kj ) then A E ar(A) . (iii) If 0 E a c(Kj) U p(Kj ) then A E ac(A), where a p(.) a c (.)a r (.) and p(.) denote respectively the point spectrum, the continuous spectrum, the residucl spectrum and the resolvent set.
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Topics in Neutron ans sport Theory
Proof: (2) Let X E u p ( A ) .Then Q = ( f o , f l , ...,f m ) l # 0 and
-kfi
+ Kifo
=
-Aj
fi
; (1
5 i 5 m).
(4.40)
The second equation of (4.40) shows that K j fo = 0. If fo = 0 then the second equation of (4.40) shows that fi = 0 for all i # j and the first equation yields f j = 0 which contradicts our assumption, whence 0 E u p ( K j ) . Conversely, let fo EKer(Kj)flD(T0) (fo # 0). We d e h e fi (i # j ) by
-kfi
+ Ki fo = -Aj fi
Kifo
i.e. fi = Xi - Xj
and f j through the equation
Hence -Aj E u p ( A ) . (ii) Let 0 E u r ( K j ) .Then 0 4 u p ( K j )and (i) implies that -Aj 4 u p ( A ) . Let ImKj = Fj be the range of K j . By assumption, it is not dense in Lp(R x V ) .We consider the problem
where Q = ( f o , f l , ..., f m ) l and @ = (go,gl,...,g m ) k One sees that (4.41) is not solvable if gj $! Fj whence -Aj E u,(A). (iii) Let 0 E a,(Kj) Up(Kj).Then 0 4 u p ( K j )and, by (i), -Aj q! u,(A). We consider the problem (4.41). The second equation for i = j reduces to K j fo = gj. Since D(To) is only dense in LP(R x V ) we can solve K j fo = gj only for g j in a dense subset of LP(R x V ) .Knowing fo we recover fi for i # j. Finally f j is deduced from the first equation. Thus the range of A-A is dense. 0
Remark 4.8 We note that the results in Theorem 4.14 do not rely on any assumption on dp or K j (0 5 j 5 m ) .
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Chapter 4. Spectral analysis. A unified theory
We study now the essential type ofthe perturbed semigroup {V(t); t ~ O} . This is necessary for the understanding of its time asymptotic structure. We denote by Rj(t) (j ~ 1) the remainder terms of the Dyson-Phillips expansion of V(t) from the unperturbed semi group {U(t); t ~ O} . Theorem 4.15 We assume that the measure dJ.L satisfies the assumption (4.15) (i. e. the Fourier transform of its truncations vanishes at infinity) and that the operators K j (0 ::; j ::; m) are regular in the sense of Definition
4.1. Then the remainder term R3(t) is compact. Consequently the essential type of {V(t);t
~
O} is equal to that of {U(t);t
~
O}.
Proof We recall that
U(t)'It =
and that the perturbation B acts as
Kofo B'It = (
+
L:
Kifo
I
)..di )
,. 'It =
Kmfo
( fo )
II
.
fm
We denote by [BU(t - s)BU(s)B'It]j (0 ::; j ::; m) the components of the vector BU(t - s)BU(s)B'It. A simple calculation shows that
[BU(t - s)BU(s)B'It]o is equal to
KoUo(t - s)KoUo(s)Kofo + KoUo(t - s)KoUo(s)f
+ where f = is equal to
L:
I
)..i(e-A,(t-s) KiUO(s)Kofo
l::1 )..di . The component
+ e-A,(t-s) KiUO(S)J)
[BU(t - s)BU(s)B'It]j (1::; j ::; m) m
KjUo(t-s )KoUo(s )Kofo+ Kpo(t-s )KoUo(s )f+
L )..ie-A,s KjUo(t-s )Kdo . i=1
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Topics in Neutron Transport Theory
We observe that, apart from the term L::l Aie-A;(t-s)KiUo(s)f, all terms involving operators of the form KaUo(r)Kv (0 ::; 0:,1/ ::; m) are shown to be compact (for r > 0) in the proof of Theorem 4.7. We consider the strong integral
We recall that, according to Corollary 4.2, the hyperplanes have zero dll'measure and consequently Ki(A - TO)-1 are compact in view of Theorem 4.1. This may be expressed as Ki : D(To)
-7
LP(f! x V) are compact (1 ::; i ::; m)
where D(To) is equipped with the graph norm. On the other hand, by a general result from semigroup theory,
Hence
r f: Aie-A;(t-s) KiUo(s)ds
Jo
is compact.
i=1
Thus i t BU(t - s)BU(s)Bds is a compact operator and consequently [U B]3 is also compact. ·T his implies the compactness of R3(t) from Theorem 2.6 and the stability of the essential type from Theorem 2.10. <> Remark 4.9 Note that the unperturbed semigroup is uncoupled so its essential type is equal to max{1],-Al, .. . ,-Am } because e-A;t (1::; i::; m) are eigenvalues of U(t) of infinite multiplicities. The assumption on the measure dJ.-L covers the continuous and the multigroup models. Remark 4.10 The spectral theory in £1 setting carries over the same lines using the Ll compactness results of the previous sections.
4.6
Comments
The material in this chapter is an expanded version of M. Mokhtar-Kharroubi [35]. Note that Theorem 4.1 was proved by V.S. Vladimirov [48] for a
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Chapter 4. Spectral analysis. A unified theory
monokinetic model by exploiting the dissipativity of the streaming operator. His argument was later used, for general continuous models by M. Borysiewicz and J. Mika [5], M. Mokhtar-Kharroubi [30] [33] and (in the context of linearized Boltzmann equation) by A. Palczewski [36]. Note that this argument is also adaptable to time-dependent problems and nonhomogeneous boundary conditions (see K. Jarmouni-Idrissi and M. MokhtarKharroubi [16]) . Compactness results for stationary transport operators, with a view to spectral problems, are also given, for instance, by S. Ukai: [44], S. Albertoni and B. Montagnini [1], J . Mika [27], E.W. Larsen and P.F. Zweifel [21]. Compactness results for multigroup models are given by K. Jorgens [17], G.H. Pimbley [37], M. Mokhtar-Kharroubi [32] . As regard to compactness for the spectral analysis of transport semigroups, besides the fundamental work of K. Jorgens [17] (when the velocity space is bounded away from zero), the basic ideas are due to I. Vidav [47] who introduced, among other ideas, continuity conditions (for the uniform topology) for operators of the form U(t1)KU(t2)K. ..u(tm)K ( ti > ,1 < i < m ) in order to recover compactness results for remainder terms of the DysonPhillips expansion (the strong convex compactness theorem (Theorem 2.2) was unknown at that time). Such continuity assumptions were somewhat neglected (except in Y. Shizuta's paper [40]) and considered difficult to satisfy for usual scattering kernels. In fact , such continuity assumptions turned out to hold for continuous models and regular collision operators (M. Mokhtar-Kharroubi [34]) . Actually they hold for much more general measures (see the proofs of Theorem 4.7 and Proposition 4.2). Compactness results for remainder terms of the Dyson-Phillips expansion are given by J . Voigt [50] who factored the second-order remainder term in order to avoid Vidav's continuity assumptions. Furthermore, J. Voigt [50] deals with possibly unbounded domains. This factorization technique was also used by M. Mokhtar-Kharroubi [33]. We point out that factoring techniques are also fully exploited by M. Borysiewicz and J . Mika [5] and Y. Shizuta [40] . Domination arguments, to prove the weak compactness in L1 of the second-order remainder term of the Dyson-Phillips expansion, were used by G. Greiner [12], J. Voigt [50] and P. Takak [43]. The use of domination arguments was systematized by M. Mokhtar-Kharroubi [33] and exploited to derive inverse compactness results. Finally, we point out the paper by 1. Weis [52] where the strong convex compactness theorem (Theorem 2.2) is given to deal with the compactness of the second-order remainder term of the Dyson-Phillips expansion thus avoiding Vidav's continuity assumptions. The spectral theory of transport generators with delayed neutrons was studied only for simple models in slab geometry by J. Mika [28] and H.G. Kaper [19] while, to our knowledge, the spectral theory of transport semigroups with delayed neutrons has never been studied. The material
°
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in Section 4.5 is an improved version of [31] Chapter VI. One of the main concerns of this chapter was an attempt to unify and extend the known compactness results. As the different direct and inverse compactness results of this chapter show, the spectral theory of T + K or et(T+K) is intimately related to the compactness of the collision operator K with respect to velocities. At this point we mention interesting open problems of physical and (or) mathematical interest. Before stating them we recall that, in solid moderators, the collision operator K is a sum of (an incoherent part) Ki and (a coherent part) Kc where Ki is compact in L2(V) while K c is not [5] . We first state an easy extension of a result by M. Borysiewicz and J . Mika [5]. Theorem 4.16 Let 1 < p < 00 and let dJ1. be a Radon measure such that the hyperplanes have zero dJ1.-measure. Let K = K1 + K2 be a collision operator such that K2 is regular. Let ,8 = inf{ a ;a>
7],
ru((A - T)-l K 1)
< 1 VA> a} .
Then (J(T + K) n {A; ReA>,8} consists of, at most, isolated eigenvalues with .finite algebraic multiplicities. Proof: Let ReA >,8. We consider the problem At.p - Tt.p - K t.p = S ,
i.e. which is equivalent to t.p - (A - T)-l K1 t.p - (). - T)-l K 2t.p = (A - T)-l S.
(4.42)
Since ru((A -T)-lK1) < 1, (4.42) is equivalent to t.p- [I
- ().. -
1 T)-1 K1r ().._T)-1 K2t.p =
[I - ().. -
1
T)-l K1r ()..-T)-lS.
From Theorem 4.1, ()..-T)-lK 2 is compact in V'(D. x V ;dxdJ1.(v)). Hence L()..) =
[I - ().. -
1
T)-l K1r ().. - T)-lK2
is a holomorphic family of compact operators. It follows from GohbergShmulyan's theorem [38] that 1- L()') is invertible except for a discrete set 3 C {A; Re).. > ,8} consisting of degenerate poles of (I - L()..))-l . Thus, for Re).. > ,8, ).. 1. 3, ().. - T - K)-l = (I - L()..))-l [I - ().. - T)-l K 1] -1
().. _
T)-l
Chapter 4. Spectral analysis. A unified theory
93
and :=: consists of isolated eigenvalues of T + K with finite multiplicities. <> We start with three problems in connection with Theorem 4.16: Problem 1: What is the structure of the spectrum of T + K in the region {TJ < ReA < .B} ? It is clear that we cannot expect interesting results in a such abstract setting. It is more useful to try concrete non-compact collision operators in the spirit of those considered by E .W. Larsen and P.F. Zweifel [21]. We point out that in £1 spaces, K 2 ()..-T)-1 is not weakly compact [13] . The weak compactness of ().. - T)-l K2 is open. Therefore the following problem seems also to be open. Problem 2: Is Theorem 4.16 true in L1 spaces? The following problem seems to be open even in LP spaces. Problem 3: Under the assumptions of Theorem 4.16, find (an estimate of) the essential type of et(T+K) . This is, of course, necessary in order to understand the time asymptotic behaviour of et(T+K) . In section 4.3, several compactness results about the remainder terms J4n(t) are given. The first one (Theorem 4.7) is based on the assumption that the Fourier transform of (the truncations of) the Radon measure dtL goes to zero at infinity. The second one (Theorem 4.8) relies on the assumption that translated hyperplanes have zero dtL-measure. Finally, the third one (Theorem 4.9), which i3 the most general, is based on the abstract assumption on dtL ensuring that the Fourier 'Transform of [d.Bt]m (for some integer m :::: 1) vanishes at infinity. Hence the natural question: Problem 4: Is it possible to translate in simple geometrical terms, as in Theorem 4.8, the condition in Theorem 4.9 ? The reader has certainly observed that the compactness results rely on different assumptions on the measure dtL depending on whether we deal with the spectral analysis of the generator T +K or the semigroup et(T+K) . Note that, for the usual Lebesgue measures on Rn or on spheres, both assumptions are satisfied. It seems that this is the reason why no counterexample to the (partial) spectral mapping theorem (4.43) has been found till now. We recall that the lack (in general) of such spectral mapping theorems was the initial motivation (see I. Vidav [47]) for studying the spectrum of the transport semigroup instead of that of its generator and was a recurrent theme in the subsequent literature on this topic. The results of this chapter seem to indicate that the first step in the direction of a possible counterexample to (4.43) should be to solve the following tricky question:
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Problem 5: Find a Radon measure dp, such that the hyperplanes have zero dp,-measure and such that, for all m ~ 1, the Fourier Transform of [d.Bt]m does not vanish at infinity. We also see that the assumptions on the measure dp, depend on whether we consider V spaces (1 < p < 00) or L1 spaces, whence the natural question: Problem 6: Is (). - T)-1 K power compact in L1 under the assumption that dp, is such that the hyperplanes have zero dp,-measure? Is it possible that, for certain measures dp" the structure of a(T+K)n(Re). > ry) depends on whether p > 1 or p = 1 ? Finally, we mention a plausible conjecture: Problem 7: Prove that the .first remainder term R 1 (t) = V(t) - U(t) is not compact (or weakly compact in L1) in general. We note that, even if the continuous and multigroup models display the same asymptotic spectral structure, they enjoy different compactness properties. Indeed, R2(t) is compact for the continuous model but probably not compact for the multigroup one (see Remark 4.7 and Proposition 4.4). This indicates that different measures dp, should induce different smoothing effects. We note also that the presence of delayed neutrons has an effect on the compactness properties since R3(t) is compact (Theorem 4.15) while R2(t) is (probably) not compact. It is to be noted that the spectral theory in unbounded domains (e.g. half-spaces or the whole space) is much less studied. However, for compactly supported cross-sections many connections exist with spectral theory in bounded domains (see A. Huber [15]). We also note that the spectral mapping theorem (4.43) holds (in an abstract setting) if t > 0 --? KU(t)K is continuous in the uniform topology (see F. Andreu, J . Martinez and J .M. Mazon [3]) and this assumption is actually satisfied by transport operators, regardless of the boundedness of n, under very general assumptions as is shown in the proofs of Theorem 4.7 and Proposition 4.2. The spectral mapping theorem (4.43) probably holds under even much more general conditions (see Conjecture 2.1) . We mention, though it is not the purpose of this book, the existence of an important spectral literature in connection with the Boltzmann equation particularly in the Japanese School (see, for instance, Y. Shizuta [40] , S. Ukai: [45], A. Palczewski [36], C. Cercignani [7] , N. Bellomo, A. Palczewski and G. Toscani [4] and references therein) . There is also a general spectral theory for one-dimensional transport operators with abstract boundary operators by K. Latrach [22] [23] [24]. In this chapter, we dealt only with the general aspects of spectral theory which are consequences of compactness results. The peripheral spectral theory, in connection with positivity, is considered in Chapter 5. Positivity also played a decisive role in the present chapter, in the analysis of compactness via domination theorems [2] [8], even if no
Chapter 4. Spectral analysis. A unified theory
95
positivity assumption is made on the collision operators. This is due to the fact that regular collision operators can be approximated by differences of positive collision operators. We will also consider much finer aspects of the point spectrum in Chapter 6 for form positive collision operators.
References [1] S. Albertoni and B. Montagnini. On the spectrum of neutron transport equation in finite bodies. J. Math. Anal. Appl. 13 (1966) 19-48. [2] D. Aliprantis and O. Burkinshaw. Positive compact operators on Banach lattices. Math. Z. 174 (1980) 289-298. [3] F. Andreu, J. Martinez and J.M. Mazon. A spectral mapping theorem for perturbed strongly continuous semigroups. Math. Ann. 291 (1991) 453-462. [4] N. Bellomo, A. Palczewski and G. Toscani. Mathematical Topics in Nonlinear Kinetic Theory. World Scientific Publishing, 1988. [5] M. Borysiewicz and J. Mika. Time behaviour of thermal neutrons in moderating media. J. Math. Anal. Appl. 26 (1969) 461-478. [6] H. Brezis. Analyse Fonctionnelle:Theorie et Applications. Masson, Paris, 1983. [7] C. Cercignani. The Boltzmann equation and its applications. Springer Verlag. Appl. Math. Sci. 67, 1988. [8] P. Doods and D.H. Fremlin. Compact operators in Banach lattices. Israel J. Math. 34 (1979) 287-320. [9] J.J. Duderstadt and W.R. Martin. Transport Theory. John Wiley & Sons, Inc, 1979. [10] N. Dunford and J.T. Schwartz. Linear Operators, Part 1. Interscience, 1958. [11] W. Greenberg, C. Van der Mee and V. Protopopescu. Boundary Value Problems in Abstract Kinetic Theory. Birkhiiuser Verlag, 1987. [12] G. Greiner. Spectral properties and asymptotic behavior of the linear transport equation. Math. Z. 185 (1984) 167-177.
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[13] F. Golse, P.L. Lions, B. Perthame and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. [14] B. Host, J.F. Mela and F. Parreau. Analyse harmonique des mesures. Asterisque, 135-136. Societe MatMmatique de France, 1986. [15] A. Huber. Spectral properties of the linear multiple scattering operator in LI-Banach lattices. Int Eq. Op Theory 6 (1983) 357-37l. [16] K. Jarmouni-Idrissi and M. Mokhtar-Kharroubi. Dissipativity of transport operators and regularity of velocity averages. Work in preparation. [17] K. Jorgens. An asymptotic expansion in the theory of neutron transport. Comm. Pure. Appl. Math. 11 (1958) 219-242. [18] H.G. Kaper, C.G. Lekkerkerker and J. Hejtmanek. Spectral Methods in Linear Transport Theory. Birkhiiuser Verlag, 1982. [19] H.G. Kaper. The initial-value transport problem for monoenergetic neutrons in an infinite slab with delayed neutron production. J. Math. Anal. Appl. 19 (1967) 207-230. [20] Y. Katznelson. An introduction to Harmonic Analysis. Dover Publication, Inc, 1976. [21] E.W . Larsen and P.F. Zweifel. On the spectrum of the linear transport operator. J. Math. Phys. 15 NOll (1974) 1987-1997. [22] K. Latrach. Theorie spectrale d'equations cinetiques. These, Universite de Franche-Comte Besanc;on, 1992. [23] K. Latrach. Compactness properties for linear transport operators with abstract boundary conditions in slab geometry. Transp . Theory. Stat. Phys. 22 (1993) 409-430. [24] K. Latrach. Quelques remarques sur Ie spectre essentiel et application a l'equation de transport. C.R. Acad. Sci. Paris. Ser J. 323 (1996) 469-474. [25] W . Littman. Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Amer. Math. Soc. 69 (1963) 766-770. [26] J .T. Marti. Mathematical foundations of kinetics in neutron transport theory. Nucleonik. 8, Bd, Helft3 (1966) 159-163.
Chapter 4. Spectral analysis. A unified theory
97
[27] J. Mika. Time dependent neutron transport in plane geometry. Nucleonik. 9, Bd, Helft4 (1967) 200-205. [28] J . Mika. The effects of delayed neutrons on the spectrum of the transport operator. Nucleonik. 9 Bd, Helft1 (1967) 46-5l. [29] B. Montagnini and M.L. Demeru. Complete continuity of the free gas scattering operator in neutron thermalization theory. J. Math. Anal. Appl. 12 (1965) 49-57. [30] M. Mokhtar-Kharroubi. La compacite dans la tMorie du transport des neutrons. C.R. Acad. Sci. Paris. Ser I. 303 (1986) 617-619. [31] M. Mokhtar-Kharroubi. Les equations de la neutronique. These d'Etat, Paris, 1987. [32] M. Mokhtar-Kharroubi. Spectral theory of the multigroup transport operator. Eur. J. Mech. B/Fluids. 9 N0 2 (1990) 197-222. [33] M. Mokhtar-Kharroubi. Time asymptotic behaviour and compactness in transport theory. Eur. J. Mech. B/Fluids. 11 n 0 1 (1992) 39-68. [34] M. Mokhtar-Kharroubi. Effets regularisants en theorie neutronique. C.R. Acad. Sci. Paris. Ser I. 309 (1990) 545-548. [35] M. Mokhtar-Kharroubi. A unified treatment of the compactness in neutron transport theory with applications to spectral theory. Publications mathematiques de Besan{:on, 1995-1996. [36] A. Palczewski. Spectral properties of the space inhomogeneous linearized Boltzmann operator. Transp. Theory Stat. Phys. 13 (1984) 409-430. [37] G.H. Pimbley. Solution of an initial value problem for the multivelocity neutron transport equation with a slab geometry. J. Math. Mech. 8 (1958) . [38] M. llibaric and I. Vidav. Analytic properties of the inverse A(z)-l of an analytic linear operator valued function A(z) . Arch. Rational Mech. Anal. 32 (1969) 298-310. [39] W. Rudin. Analyse reelle et complexe. Masson, Paris, 1978. [40] Y. Shizuta. On the classical solutions of the Boltzmann equation. Comm. Pure. Appl. Math. 36 (1983) 705-754.
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[41] E.M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971. [42] S. Steinberg. Meromorphic families of compact operators. Arch. Rational Meeh. Anal. 31 (1969) 372-379. [43] P. Takak. A spectral mapping theorem for the exponential function in linear transport theory. Transp . Theory Stat. Phys. 14(5) (1985) 655-667. [44] S. ukai. Eigenvalues of the neutron transport operator for a homogeneous finite moderator. J. Math. Anal. Appl. 30 (1967) 297-314. [45] S. Ukai. Solutions of the Boltzmann equation, in Patterns and wavesQualitative analysis of nonlinear differential equations, pp. 37-96. Studies. Math. Appl. 18, 1986. [46] I. Vidav. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22 (1968) 144-155. [47] I. Vidav. Spectra of perturbed semigoups with applications to transport theory. J. Math. Anal. Appl. 30 (1970) 264-279. [48] V.S. Vladimirov. Mathematical Problems in the One-velocity Theory of Particle Transport. Atomic Energy of Canada. Ltd. Chalk lliver. Ont Report. AECL-1661 (1963). [49] J. Voigt. Positivity in time dependent linear transport theory. Acta. Appl. Math. 2 (1984) 311-331. [50] J . Voigt. Spectral properties of the neutron transport equation. J . Math. Anal. Appl. 106 (1985) 140-153. [51] J. Voigt. Functional analytic treatment of the initial boundary value problem for collisionless gases. Habilitationschrijt, Universitiit Miinchen, 1981. [52] L. Weis. A Generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129 (1988) 6-23. [53] K. Yosida. Functional Analysis. Springer Verlag, 1978.
Chapter 5
On the leading eigenelements of transport operators 5.1
Introduction
This chapter is devoted to a thorough analysis of the leading eigenelements of neutron transport operators. It is well known that they are the only eigenelements of physical significance and their importance in nuclear reactor theory (e.g. pulsed neutron experiments) motivated a great deal of the subsequent spectral literature in neutron transport theory. The existence of such leading eigenelements is strongly tied to positivity. The role of positivity in nuclear reactor theory was emphasized very early by G. Birkhoff [7] [8] [9] and G.J. Habetler and M.A. Martino [21]. Basic results on the leading eigenelements of transport operators were given by I. Vidav [45], J. Mika [32], T. Hiraoka and S. Uka'i [22], N. Angelescu and V. Protopopescu [3] and other developments followed. On the other hand, independently of transport theory, the spectral theory of positive operators and positive semigroups developed in its own right to a high degree of refinement and now very general functional analytic results are available which can be found , for instance, in the monograph by R. Nagel et al [38]. In Section 5.2, we recall some of the basic results on the spectral theory of positive operators. The aim is not, of course, to exhaust the theory but simply to select those results which present a great interest for neutron transport problems. Besides relatively well-known results on the peripheral spectrum, we will 99
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present more specialized results such as strict comparison of spectral radii by I. Marek [30],the superconvexity result by J.F.C. Kingman [27] and T. Kato [26], B.de. Pagter's result on irreducible compact operators [40] and related results. We will show, in the subsequent sections, how such tools illuminate the properties of the boundary spectrum of transport operators. For a quick introduction to basic definitions and spectral results on positive semigroups on Banach lattices we refer to Ph. Clement et a1 [15]. A more systematic account is given in R. Nagel et a1 [38]. In Section 5.3, we present several approaches of the irreducibility of transport semigroups giving different points of view on this problem. In Section 5.4, we give very general existence results for eigenvalues, when the velocity space is bounded away from zero. Such results rely on the compactness results of Chapter 4 and on irreducibility arguments. In Section 5.5, we derive a spectral link between the transport operator and its bounded part. This link provides us with nonexzstence results we give in Section 5.6. We also give, in this section, an upper bound of the leading eigenvalue in terms of the spectral radius of the collision operator, when the velocity space is bounded away from zero. In Section 5.7, we study the case where the velocity space is not bounded away from zero and give several existence results for eigenvalues, based on estimates from below of the spectral radii of suitable operators. Section 5.8 is devoted to strict monotonicity results of the leading eigenvalue with respect to the cross sections or the spatial domain. In Section 5.9, we give a result on the domain derivative (with respect to vector fields) of the leading eigenvalue for a model transport operator and a simple formula for the derivative. We sketch, in Section 5.10, an approximation theory (with error estimates) of the leading eigenelements of transport operators, based on a projection method. The criticality eigenvalue problem is dealt with in Section 5.11. Finally, Section 5.12 is devoted to the extension of (some of) the previous results to transport equations with delayed neutrons.
5.2
Spectral properties of positive operators
Let X be complex Banach lattice. We denote by X+ the cone of positive elements. The notation x > 0 means x E X+ and x # 0. We also define the dual cone by X' - x ' E x ' ; ( x ' , x ) ~ o V x e X +
+-I
1
where X' is the dual space and (., .) is the duality pairing. In the Lp(R; dp) spaces those definitions correspond to the usual notion of non-negative functions. An operator 0 E L(X) is said to be positive if it leaves the positive cone invariant. We recall that a co-semigroup {U(t); t 2 0) on X with gen-
101
Chapter 5. On the leading eigenelements
erator T is said to be positive if each operator U (t) is positive. We point out the following useful characterization Proposition 5.1 ([15] p . 161). {U(t); t ~ O} is positive if and only if (AT)-l is positive for some A > w, where W is the type of {U(t); t ~ O}.
For a general CO-sernigroup {U(t); t that (A - T)-l =
~
O} in a Banach space, it is known
l~o e->.tU(t)dt
where W is the type of {U(t); t convergent . We recall that
~
; ReA>
(5.1)
W
O} . The integral (5.1) is absolutely norm
s(T):= sup ReA::;
W
>'E<7(T)
and that the equality does not hold in general, even for positive sernigroups (see [38] p. 61) . For positive semigroups (5.1) extends in the form Proposition 5.2 ([15] p. 164) . Let{U(t);t on X with generator T. Then
1
~
O} be apositiveco-semigroup
00
(A - T)-l =
e->'tU(t)dt ; ReA> seT)
(5.2)
where (5.2) exists as a norm convergent improper integral.
The concept of irreducibility is probably one of the most inIportant ones in the theory of positive operators. To introduce it, we recall that a subspace Y of X is said to be an ideal if Ixl ::; Iyl and y E Y inIply x E Y where II denotes the absolute value. In the V(n ;dJ-L) (1::; p < 00) spaces, the ideals are of the form
{c,o
E
£P(n; dJ-L); c,o = 0 on A} for some measurable subset A
c n.
A positive operator 0 E L(X) is said to be irreducible if there is no closed ideal (except X and {O}) which is invariant under O. The irreducibility can be characterized in the following way. Proposition 5.3 A positive operator 0 E L(X) is irreducible if for any x > 0 and Xl > 0 there exists an integer n (depending possibly on x and Xl) / such that (onx,x ) > O.
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An element x E X+ is called quasi-interior if (x', x) > 0 for any x' > o. A functional x' E X~ is called strictly positive if (x', x) > 0 for any x > o. In the V(n; dft) spaces, quasi-interior elements correspond to strictly positive almost everywhere functions. A positive operator 0 E L(X) is called strongly irreducible (or positivity improving) if Ox is quasi-interior for any x > O. We note that 0 is irreducible if some power of 0 is positivity improving. This provides us with a practical mean to check the irreducibility. A positive CO-semigroup {U(t); t ~ O} on X is said to be irreducible if there is no closed ideal (except X and {O} which is invariant under U (t) for all t ~ O. We point out that a sufficient (but not necessary) condition for a positive CO-semigroup {U(t); t ~ O} to be irreducible is that, for some to > 0, the operator U(to) be irreducible. The following characterization proves useful for the sequel
r
Proposition 5.4 ([15] p. 165). Let {U(t); t ~ O} be a positive co-semigroup on X with generator T. Then the following assertions are equivalent: (i) {U(t); t ~ O} is irreducible. (ii) For every x> 0, x' > 0 there exists t ~ 0 such that (U(t)x, x') > O. (iii) (A - T)-l is strongly irreducible for all (for some) A > s(T). (iiii) (A - T)-l is irreducible for all (for some) A > s(T). We are now in a position to give the main spectral results. We begin with the most important one Theorem 5.1 ([41]). Let 0 E L(X) be a positive operator. Then ra(O) E a(O) . The analog for generators of positive semigroups is the following Theorem 5.2 ([15] p. 202). Let {U(t); t ~ O} be a positive co-semigroup on X with generator T. Then the spectral bound of T s(T)
=
sup ReA E a(T) AEa(T)
provided that a(T)
=1=
0.
The peripheral spectrum of a positive operator 0 E L(X) is defined by
{A E a(O); IAI = ra(O)} while the peripheral (or rather the boundary) spectrum of the generator T of a positive co-semigroup is defined by a+(T) = {A E a(T); ReA = s(T)} .
The peripheral spectrum enjoys nice properties
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Chapter 5. On the leading eigenelements
Theorem 5.3 ([15] pp. 202 and 205). Let {U(t)j t 2: O} be a positive cosemigroup on X with generatorT. If s(T) is a pole of the resolvent ().._T)-l of order p, then any other pole J1, E O'+(T) is of order::; p. Moreover O'+(T) is cyclic, i. e. ).. E 0'+ (T) implies s(T) + ikhn)" E 0'+ (T) for every k E Z . More precise pictures of O'+(T) hold under additional assumptions
Theorem 5.4 ([15] p. 209). Let {U(t)j t 2: O} be an irreducible co-semigroup on X with generator T. Then there exists II 2: 0 such that O'+(T) = s(T)
+ illZ
and the elements of O'+(T) are first order poles with algebraic multiplicity one. Moreover, there exists a quasi-interior Xo E X+ such that Txo s(T)xo and a strictly positive x~ E X~ such that T' x~ = s(T)x~. Replacing the irreducibility condition by an assumption on the essential type yields
Theorem 5.5 Let {U(t)j t 2: O} be a positive co-semigroup on X with generator T. We assume that We < W where wand We are respectively the type and the essential type of {U(t)j t 2: O} . Then
Furthermore, there exists c: > 0 such that O'(T) n {Aj Re).. 2: W i.e.
W
-
c:}
= {w}
is a strictly dominant eigenvalue ofT.
Proof: The result follows easily from the fact that O'(T) n {>.; Re).. 2: Q} is finite for We < Q < W and from the cyclicity of the boundary spectrum (Theorem 5.3) . <> We mention a useful tool to check the irreducibility for positively perturbed positive CO-semigroups which can be used for transport problems
Theorem 5.6 ([38] p . 307). Let {U(t)j t 2: O} be a positive co-semigroup on X with generator T and let K E L(X) be positive. Let {V(t)j t 2: O} be the co-semigroup generated by T + K. Let I c X be a closed ideal. Then the following assertions are equivalent (i) I is invariant under {V(t)j t 2: O} . (ii) I is invariant under both {U(t)j t 2: O} and K.
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An important question of pure and applied interest is under which conditions is the spectral radius of a positive operator strictly positive ? The known criteria (up to 1983) (including Ando-Krieger's theorem for integral operators) tied to the concept of irreducibility are summarized in H.H. Schaefer ([42] Theorem A). We point out an extension of Ando-Krieger's theorem to ordered Banach spaces by V. Caselles [12] . We mention here a general abstract result by B.de. Pagter [40] (published in 1986) which will be used quite often in the sequel Theorem 5.7 Let 0 E L(X) be a compact irreducible compact operator. Then ru(O) > O.
The following result, due essentially to H.H. Schaefer [42] , will prove useful for transport problems Theorem 5.8 Let {U(t) j t ~ O} be an irreducible eo-semigroup on a Banach lattice X with generator T If U(t) is compact for t large enough then
a(T)
-10.
Actually this result is not stated in Schaefer's paper [42] . However its proof is exactly the same as that of the case X = L1(p,) ([42] Theorem B (iii)) by using B.de. Pagter's result [40]. The previous result provides us with the existence of an eigenvalue for generators under certain conditions on the semigroups. We will give thereafter (Theorem 5.12) another approach (for perturbed semigroups) tied to properties of the generator and also well suited to transport problems. To this end we introduce the concept of superconvexity and its applications to positive operators depending on a real parameter. This concept also plays a basic role for an approximation theory of the leading eigenelements of transport operators [13] (see also Section 5.10) . Definition 5.1 Let I be an interval of R . A function f : I superconvex if Log f(x) is a convex function.
-t
R+ is called
We first mention a result on the superconvexity of the spectral radius of positive matrices by J .F .C. Kingman [27] Theorem 5.9 Let A(h) = (ai ,j(h))~j=1 be a positive matrix depending on a parameter h E I . If ai,j (h) is superconvex for each i , j E {I, .. ., n}, then
h is superconvex.
-t
ru(A(h))
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Chapter 5. On the leading eigenelements
We present now an infinite dimensional version of J.F.C. Kingman's result due to T. Kato [26] . To this end we recall the meaning of the superconvexity for vector valued mappings. Let A(h) be a family (indexed by hE 1) of operators acting on a Banach lattice X with positive cone X+. A function
f: hE I
-->
f(h) E X+
is called superconvex if, for each E: > 0 and each triplet hI ::; ho ::; h2 , there are finitely many Xj E X+ and real superconvex functions cpj(h) such that
f(hk) -
L CPj(hk)Xj
::;
E:
;
k = 0, 1,2.
j
We say that
A: hE 1--> A(h) E L(X) ; A(h)
~
0
is superconvex if, for each x E X+, hE 1--> A(h)x E X+ is super convex. We state T. Kato's result [26]
Theorem 5.10 If A(h) is superconvex in h, so is rO'(A(h)). Actually for the purpose of Theorem 5.12 we give a particular and simpler version of T. Kato's result based on the more tractable concept of weak superconvexity. A family A(h) E L(X) ; A(h) ~ 0 is called weakly superconvex if the real valued function (A(h)x, x'} is superconvex for each x E X+ and x' E X~ . Then we have the following result which can be deduced directly from J.F.C. Kingman's result by approximation:
Theorem 5.11 ([13]). Let A(h) E L(X); A(h) ~ 0 be a weakly superconvex family of compact operators. Then rO'(A(h)) is superconvex. We are in a position to prove a result complementing Theorem 5.8:
Theorem 5.12 Let {U(t); t ~ O} be a positive co-semigroup on a Banach lattice X with generator T. Let there exist a > 0 such that U(t) = 0 for t > a. Let B E L(X) be a positive operator and {V(t); t ~ O} be the cosemigroup generated by T + B. If some power [(>' - T)-I B]m (>. E R) is compact and irreducible then a(T + B) i- 0.
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Topics in Neutron lkansport Theory
Proof: We give two proofs of this results. We observe that (A - T)-' is entire since {U(t);t 0) is nilpotent. The first proof is based on the assumption (different from that of the statement) that (A-T) -' [B(X - T) -'1" is compact and irreducible for some integer m. The second proof (corresponding to the statement) relies on Kingman-Kato's convexity result. Let s(T B) be the spectral bound of T B
>
+
+
It is known (see [38] Proposition 2.5, p. 67) that, for X r, [(A - T -
B)-'1
=
1 dist(X;u(T
+
T)-'1
1
-
+ B)) - X - s(T + B)
so that u(T B) # 0 if and only if r, [(A - T hand, according to Lemma 8.1, r, [B(X-
> s(T + B),
B)-'1
> 0. On the other
< 1 for X > s(T + B)
and
Hence
from Theorem 5.7 and this ends the first proof. We give now a differenta p proach. By analyticity arguments, [(A - T ) - ' B ] ~is compact for all X and, by Gohberg-Shmulyan's theorem, u(T B) consists, at most, of isolated eigenvalues with finite algebraic multiplicities. In view of Theorem 5.2, u(T + B) # 0 is equivalent to the existence of a real eigenvalue. According to Theorem 5.7 and the spectral mapping theorem, r,((X - T)-'B) > 0 is an eigenvalue of (A - T)-'B associated with a positive eigenfunction. On the other hand, X is an eigenvalue of T B if and only if 1 is an eigenvalue of (A - T)-'B. Hence the problem amounts to the existence of X E R such that r,((X - T)-'B) = 1. By analyticity arguments,
+
+
X E R + r,((X
- T)-'B)
whence the problem amounts to
is strictly decreasing
107
Chapter 5. On the leading eigenelements We start with
Set 0(>')
= [(>. - T)-1 B]m >. E R
---->
and let us show that
0(>') is weakly superconvex,
i.e. for all x> 0 and x' > 0,
is superconvex. The integrand is superconvex. Since linear combinations with positive coefficients and limits of superconvex functions are superconvex ([26] Lemma 2.2), then (5.4) follows by approximating the integral by finite sums and passing to the limit. According to Theorem 5.11,
>. E R
---->
ru(O(>'))
is super convex and hence convex. We note that ru(O(>')) is analytic in >. because ru(O(>')) is a simple eigenvalue of the analytic operator 0(>') (see T. Kato [25]). Finally, as a differentiable strictly decreasing convex function, lim ru(O(>')) = .:\.-+-00
+00
so (5 .3) is satisfied. <> Another important question of pure and applied interest concerns the strict comparison of the spectral radii of two operators in a context of domination. Such questions were investigated a long time ago by 1. Marek [30]. For more recent results and references we refer to V. Caselles [11] . We quote here a very particular version of a result by 1. Marek [30] which will be used later to study the strict monotonicity of the leading eigenvalue of transport operators with respect to various parameters. Theorem 5.13 Let 0 1 and 02 be two positive bounded operators such that
and let 02 be power compact and irreducible. Then r u (01)
< ru(02) .
108
Topics in Neutron ~ m s p o r Theory t
The irreducibility of transport semigroups
5.3
Let R be a smooth open subset of Rn and let dp be a positive Radon measure on Rn such that dp{O) = 0. We denote by V the support of dp and refer to V as the velocity space. We consider neutron transport equations of the form
a f v.-a f at ax
+
+ ~ ( xv ),f ( x ,v , t )
J
=
k ( z ,v , v ' )f
( 2 ,v', t ) d P ( v t )
where
r- = { ( x , ~E)dR x
V ; v.n(x) < 0 )
and n ( x ) is the outward normal at x E aR. Here, u ( x , v ) denotes the collision frequency at x for neutrons with velocity v , while k(., ., .) is the scattering kernel. We recall the streaming operator
T f = -v.-
af - ~ ( xv ),f ( x ,v ) ; f E D ( T )
ax
with domain
f E L p ( R x V ) ; v.-a f E Lp(R x V ) ,fir- = O
ax
where
Lp(R x V ) = L p ( R x V ; d x d p ( v ) ) ; 1 I p < oo. We assume, as usual, that the collision frequency is non-negative and bounded and that the scattering kernel is non-negative and defines a bounded operator K on P ( R x V). Note that T generates the following Q-semigroup
where
s(x,v)=inf{s>O;x-sv$R). This section is devoted to the irreducibility of the %-semigroup { V ( t ); t 2 0 ) with generator T K. Different approaches of the irreducibility are given. We begin with a result by J. Voigt [46] concerning the continuow models.
+
109
Chapter 5. On the leading eigenelements
We first recall the notion of path diameter p-diam( n), for an open connected n c Rn. For x, x' E n, we define the path distance p-dist(x, x') = inf {length of C; C polygonal path connecting x and x'} p-diam(n) = sup {p-dist(x, x'); x, x'
En} .
Theorem 5.14 Let n be connected and let VeRn (open) and df-£(v) the restriction of the Lebesgue measure to V . Let there exist 0 ~ Cl < C2 ~ 00 such that
and k(x, v, v') > 0 a.e. on (n x Vo x V) u (n x V x Vo)
(5.7)
then {Vet); t ;::: O} is irreducible. Moreover, if C2 = 00 then {Vet); t ;::: O} is positivity improving for all t > O. If C2 < 00 and p-diam(n)< 00 then {Vet); t ;::: O} is positivity improving for all t > P-di~:,,(n). The proof, of geometrical character, is based on Proposition 5.4 part (ii) and the analysis of the second term of the Dyson-Phillips expansion of {Vet); t ;::: O} from {U(t); t ;::: O}. We present now a second approach of the irreducibility based on the resolvent characterization (Proposition 5.4 part (iii)) . Theorem 5.15 Let there exists an integerm such that [()._T)-lK]m is positivity improving, then {Vet); t ;::: O} is irreducible. This condition is satisfied (with m = 2) by the continuous models under the assumption (5.7) and the additional assumption that n be convex.
Proof: For)' large enough, 00
(). - T - K)-l =
L [(). - T)-l K]i (). - T)-l o
and then (). - T - K)-l is irreducible if some power of (). - T)-l K is positivity improving. For the Lebesgue measure on Rn, K()' - T)-lK is an integral operator whose kernel is greater than or equal to
x(x-x
I
' I <sex,
x- x 10 e- ().+-(<7-,z-z ))t ' X - x" X - x dt -1--'1)) k(x'-t-,v)k(x,v,--)-n x- x 0 t t I
00
I
'
I
no
Topics in Neutron Transport Theory
where 0'( v)
1
= infxE!1 a(x, v) . Under
00 _(A+U(X-X'
o
e
t
»t k (x
I
.
Assumption (5.7),
X - x' ") ( x - x' )dt 0 - - v k x v - - - n > a.e.
"t
t'
t
On the other hand, I
1:=:/1))=1
X(lx-X'I<s(x,
if and only if n is convex. This ends the proof.
VX,X/En
0
Remark 5.1 We point out that the remainder terms of the Dyson-Phillips through the Laplace expansion are related to the powers [(A - T)-l transform so that it is not surprising that similar assumptions on the scattering kernel appear in both approaches. However, similar arguments led G. Greiner [18] [19] to a quite different criterion: k(x, v, v') > 0 in a neighborhood of the boundary
Kf
an.
Theorem 5.15 admits a multigroup version we give now. To this end it is convenient to recall the generation theory in this special context. The multigroup transport equation has the following form (5.8)
subject to initial and boundary conditions
where Here 1f;i(X, v, t), with (x, v, t) E particles with speed Ci. Finally
fi
n x Vi
= {(x,v) E
x R+, denotes the distribution of
an x Vi ;v.n(x) < O} .
We write (5.8) formally as a Cauchy problem d1f;
di"= T1f; + K1f;
; 1f;(O) = 1f;o
111
Chapter 5. On the leading eigenelements
T-
0 T2 0 0
C' 0 0 0
0 0 0
0 0 0 Tm
)
at.p ; Tit.p = -v. ax - (Ji(X, v)t.p(x, v) , t.p E D(Ti)
and
The entries of the matrix collision operator K = {Ki,j} (1 :s; i,j ::; m) consist of integral operators (with respect to velocities) with kernels ki ,j(x, v, v') ~ O. We note that Ti generates the eo-semigroup
Ui(t)t.p = e
-it
u,(x-rv,v)dr
t.p(x - tv, v)x(t
0
< sex, v)) ; t.p
E p(n x Vi).
Thus the diagonal operator T with domain D(T) =
II
D(Ti)
l$i$m
generates a eo-semigroup {U(t); t ~ O} on
ITl$i$m
£P(n x Vi) where
Assuming that Ki,j E L(LP(n x Vj), p(n x Vi)) ; (1::; i,j
:s; m) ,
it follows that the Cauchy problem (5.8) is well-posed and governed by a multigroup transport semigroup {Vet); t ~ O} . We are now ready to give an irreducibility criterion for the multigroup transport semigroup. Theorem 5.16 Let n be convex. We assume that, for each 1 ::; i,j ::; m, there exists 1 :s; e :s; m such that ki,e(x,v,v')
> 0 on n x Vi
X
Ve ; ke,j(x,v',v)
> 0 on n x Ve x Vj (5.9)
then the multigroup transport semigroup is irreducible.
112
Topics in Neutron Transport Theory
Proof A calculation shows that the entries of the matrix K()" - T)-l K are given by [K(). - T)-l KL,j =
L
Ki,e(). - Te)-l Ke,j
l:5e:5m
and that Ki,e(). - Te)-l Ke,j E L(V(n x Vj), V(n x Vi)) is an integral operator with kernel greater than or equal to Ni,e,j (x , x' , V , v")
I
x-x )ke J. ( X = Cen-2ki e ( x, v, ce f:-:7T' I ,
IX-X
J
I
I"
, ce
x- x ,v f:-:7T' I
)
IX- X
where O'e(v) = infxEf! ae(x, v) . Assumption (5.9) is sufficient for .. [K()' - T)-l K] ',J
to be positivity improving and it follows that [(). - T)-l K]2 is also positivity improving. We conclude by using the abstract result given in Theorem 5.15. 0 Remark 5.2 The irreducibility of the transport semigroup may also be checked by means of Theorem 5.6. We refer to [38] p. 309, for the analysis of the one dimensional case.
5.4
A general existence result
We turn to the general operator we started with in the previous section A=T+K
where
8f Tf = -v . 8x - a(x, v)f(x, v)
f E D(T)
with domain D(T)
and Krp =
=
{f E V(n x V;dxdJ.L(v));
J
v.:~
E
LP, fw- =
o}
k(x,v,v')rp(x ,v' )dJ.L(v') ; rp E V(n x V;dxdJ.L(v)).
Chapter 5. On the leading eigenelements
113
We prove now the existence of an eigenvalue for T + K, when the velocity space is bounded away from zero, under very general assumptions. In such a case, the unperturbed semigroup {U(t); t ~ O} is nilpotent, i.e. U(t) = 0 for t > to = ..4. where d is the diameter of n and Vo is the minimum speed. Vo We give two general existence results based respectively on two different functional analytic arguments (Theorem 5.8 and Theorem 5.12). Theorem 5.17 Let 0 rf. V and the transport semigroup {V(t); t ~ O} be irreducible. We assume that the collision operator is regular (in the sense of Definition 4.1) and that n is bounded. If 1 < p < 00 and if the Radon measure d/l is such that translated hyperplanes have zero d/l- measure, then a(T+K) i- 0. If p = 1 and if the Radon measure d/l satisfies the geometrical condition (4.31), then a(T + K) i- 0.
Proof: When 1 < p < 00 then, according to Theorem 4.8, the remainder R3 (t) of the Dyson-Phillips expansion is compact in V (n xV; dxd/l( v)) . When p = 1 then, according to Theorem 4.12, R3(t) is weakly compact in Ll(n x V; dxd/l(v)).1t follows, from Corollary 2.1, that R7(t) is compact in Ll(n x V; dxd/l(v)). Hence, in both cases, some remainder term is compact. On the other hand, U(t) = 0 for t > ~ = to and a simple calculation shows that V(t) = Rm(t) for t > mto. Thus V(t) is compact for t large enough and the result follows from Theorem 5.8. 0 Remark 5.3 It is possible to improve further the previous result. The irreducibility relies on some strict positivity assumption on the scattering kernel as is shown in the previous section. Actually, it suffices that, for some ball Ben, the transport semigroup corresponding to the ball be irreducible since it is easy to see that the leading eigenvalue increases with the domain. Thus, in the context of continuous (resp. multigroup) models, it suffices that the condition (5.7) (resp. (5.9)) be satisfied in B instead of
n. Theorem 5.18 Let 0 rf. V and let n be bounded. We assume that the collision operator is regular (in the sense of Definition 4.1). (i) Let 1 < p < 00 and assume that d/l is such that the hyperplanes have zero d/l-measure. If K().. - T)-l is irreducible then a(T + K) i- 0. (ii) Let p = 1 and assume that d/l satisfies the geometrical condition (4 .9) . If [K().. - T)-1]2 is irreducible then a(T + K) i- 0.
Proof: According to Theorem 4.1 (resp. Theorem 4.4), K()" - T)-l (resp. [K()" - T)-1]2) is compact in V(n x V;dxd/l(v)) (resp. in Ll(n x V; dxd/l(v))) and the proof follows from Theorem 5.12. 0
Topics in Neutron Transport Theory
114
Remark 5.4 In the context of continuous or multigroup models (and if 0 is convex), the conditions (5.7) or (5.9) ensure the irreducibility of [K(A - T)-l]2 and thus the conclusion of Theorem 5.18. As pointed out in Remark 5.3, it suffices that such conditions be satisfied in a ball B c O.
5.5
A spectral inequality
We consider, in this section, the case where the velocity space is not bounded away from zero. We will assume that the collision frequency, the scattering kernel are homogeneous and that lim inf a(v) v-+O
= vEV inf a(v) = 'T} .
We are going to show a useful link between the spectrum of the transport operator and that of its bounded part B : r.p E P(Vj dJL(v))
~ -a(v)r.p(v) +
J
k(v, v')r.p(v')dJL(v') E P(Vj dJL(v)).
We begin with the following observation Proposition 5.5 Let K be power compact in V(V; dJL(v)) . Then aas(B) := a(B)
n p;ReA >
-'T}}
consists of, at most, isolated eigenvalues with finite algebraic multiplicities. Moreover if aas(B) =/:- 0 then there exists a leading eigenvalue :X. Proof: The resolvent of the multiplication operator r.p E P(V; dJL(v)) ~ -a(v)r.p(v)
is given by r.p(v) r.p E LP(V; dJL(v)) ~ A + a(v) = (A
+ a)
-1
r.p ; ReA>
-'T}.
A!
For real A, K(A + a)-l is dominated by K and is thus power compact. This implies the first part of the propositiori. The rest follows from general results on positive operators, for instance Theorem 5.2. <> We are in a position to state Theorem 5.19 Let aas(T + K) = aCT + K) n {A; ReA> -'T}} =/:- 0 and let A be the leading eigenvalue ofT + K. Then aas(B) =/:- 0 and A ::;:X. If K is irreducible in V(V ;dJL(v)) then A
<:x.
115
Chapter 5. On the leading eigenelements Proof: We recall that
(A - T)-lc.p
r(x,V)
= Jo
e-(·X+u(v»c.p(x - sv, v)ds ; c.p E LP(n x V; dxdJ.L(v))
and it is easy to see that
Tc.p + K c.p
= Ac.p; c.p ?: 0
is equivalent to i.e. ( ,(x,V)
Jo
e-(A+u(v»ds
J
k(v, v')c.p(x - sv, v')dJ.L(v') = c.p(x, v).
(5.10)
Let d be the diameter of n. We define
1j;(.) =
In c.p(y, .)dy
Then, integrating (5.10) over
n,
...!L
J Ivl
1j;(v)
<
E LP(V; dJ.L(v)).
e-(A+u(v»ds
J
k(v, v')1j;(v')dJ.L(v')
o
where
k(A,d,v,v') =
l
d
TUf
e-(>,+u(v»ds x k(v,v').
The inequality (5.11) implies
Clearly
K(d, A) where
:s: K(oo, A)
and K(d, A) =I- K(oo, A)
-
K(OO,A) : c.p E LP(V;dJ.L(v))
1
00..0.7
A+a(v)Kc.p.
(5.12)
Topics in Neutron Transport Theory
116
Hence ru(K(oo, ,\)) 2: ru(K(d, ,\)) 2: 1 and, since A>
-7] --+
ru(K(oo, A))
is continuous (because K(oo, A) is power compact and ru(K(oo, ,\)) is an eigenvalue), nonincreasing and tends to zero at infinity, there exists); 2: A such that ru(K(oo, );)) = l. Thus there exists a non-negative g such that K(oo, );)g = g ,
i.e.
1 =----Kg=g
A+cr(V)
which amounts to Bg = );g. In the case where K is irreducible in V(Vj dJl(v)) then K(oo, A) is also irreducible and, thanks to Theorem 5.13, (5.12) yields ru(K(oo, A)) > ru(K(d, A)) 2: 1
which implies that A < );.
5.6
0
Nonexistence results
We give some consequences of the previous section. We begin with Corollary 5.1 Let K be quasinilpotent (i.e. ru(K) = 0) thenaas(T+K) =
o regardless of the size of n.
Proof: Indeed, we have K(oo,);) ~ >:!ryK and consequently the equality ru(K(oo, );)) = 1 is not possible.
0
Remark 5.5 If the scattering kernel is such that k(v, Vi) = 0 for
Ivl 2:
Iv'l
then ru(K) = O. It is possible to exploit Theorem 5.19 to derive nonexistence results of a different kind. For instance
Theorem 5.20 Let p = 1 and let a(v) >
7]
a.e. We assume that
k(v, Vi) dJl( v) ~ l. a () v -7]
J
Then aas(T + K) = 0 regardless of the size of n.
(5.13)
117
Chapter 5. On the leading eigenelements
Proof: It suffices to show that (5.13) implies that <Jas(B) = 0. Suppose the contrary and let>: be the leading eigenvalue of B. Thus 1 A + <J( v)
J ' , , -
1 k(v, v )rp(v )dll(V ) = rp ; A> -1/ , rp E L+(V ; dll(V )) , rp
f:: O.
By integrating over V we get
Hence
which ends the proof. <> Remark 5.6 If 1/ > 0 then similar calculations show that T subcritical (i .e. <Jas(T + K)
c {ReA < O} V D)
sufficient condition of subcriticality in L2 is /
if
J
+K
is always
kS'(':;)dll(V) :S 1. A
Ja~~);;l~)dll(V)dll(V') :s:
1.
We give another remark Remark 5.7 Eigenvalues cannot exist if the velocity space V does not contain at least a subset symmetric with respect to the origin. Indeed, if V is a half ball then ra((A - T)-l K) = 0 (see [36]). Although unphysical, this example shows that the converse to Theorem 5.1 9 is not true in general since the spectral theory of B is not related to the geometry of the velocity space v.
We mention now the classical disappearance phenomenon for small bodies Theorem 5.21 Let K be the closed operator in V(V; dll(V)) K: rp E D(K) { D(K)
-->
Krp =
Ivl- 1 Krp
= {rp E V(V; dll(V)); Ivl- 1 K rp E V(V ; dll(V)) } .
If D(K) = V(V; dll(V)) then <Jas(T+K) diameter of D.
= 0 for d IIKII :S 1 where d is the
Topics in Neutron Transport Theory
118 Proof: We note that
(A - T)-l'tjJ =
Ivl
_
( S(X,w)
1
io
_
e
v
( A+U ( V )) '
Ivi
'tjJ(x - sw, v)ds ; w =
f0
and that
Illvl (A - T)-lIIL(LP(nXV;dXdJ.L(V») ~ d. Hence
II(A - T)-l KII < d IIKII ; ReA> -TJ
which ends the proof since 1 ~ ap((A - T)-lK) . 0 We end this section with an upper estimate of the leading eigenvalue when the velocity space is bounded away from zero. Theorem 5.22 Let 0 ~ V, let c be the minimum speed and d be the diameter of n. If a(T + K) #- 0 then the leading eigenvalue A of T + K satisfies the estimate d
la c e-(>.+infu(,))sds x ru(K) ~ 1.
(5.14)
Proof. Now, for any A, r(X,V)
(A - T)-lcp =
io
e-(>'+u(v»cp(x - sv , v)ds ; cp E LP(n x V; dxd{t(v)).
By using the calculations in Section 5.5, (5.11) shows that d
'tjJ(v)
~ J~VI ~
J:
e-(>'+u(v»ds
J k(v,v')'tjJ(v')d{t(v')
d
e-(>.+infu(,»sds x K'tjJ =: K'tjJ
so that ru(K) ~ 1 and this proves the claim.
5.7
0
Existence results
We deal, here, with the case where the velocity space is not bounded away from zero. We also assume that the collision frequency, the scattering kernel are homogeneous and that lim inf a(v) = inf a(v) = TJ . V""" 0
vEV
119
Chapter 5. On the leading eigenelements We assume that
(>. - T)-l K is power compact in V(n x Vj dxd/L(v)) .
(5.15)
Conditions on K implying (5.15) are given in Chapter 4. To fix the ideas we will assume that K is compact on V(Vj d/L( v)) (1 < p < (0) or dominated by a compact operator in L1(Vjd/L(v)). If (>. - T)-lK is irreducible then Theorem 5.7 ensures that
ra«>' - T)-l K) > O. By analyticity arguments (see Gohberg-Shmulyan's theorem [43]) ,
>.
]-7], oo[
E
-+
ra«>' - T)-l K)
is strictly decreasing (and continuous). It follows that
if and only if lim ra«>' - T)-l K) > 1
>'-+-'7
and then the leading eigenvalue>: of T
+K
is characterized by the equality
Thus, we are faced with the problem of obtaining explicit lower bounds of the spectral radius ra«>' - T)-l K) or, which amounts to the same, explicit lower bounds of ra(K(>. - T)-l) . We set r(X,V')
a(x , v', >.)
= Jo
,
e-(A+a(v))sds
j
>. 2: -7].
n x ]-7],00[, we define the operator
For each (x, >.) E
H(x, >.) : t.p
E
V(Vj d/L(v))
-+
J
a(x, v' , >')k(v, v')t.p(v')d/L(v').
It is easy to see that H( x, >.) inherits the compactness assumptions of K . We are going to define a parameter T(>') playing the role of a "uniform spectral radius" of H(x, >.), i.e. independent of x E n. We proceed as follows. Let
X(>.) =
b
2: OJ :3 t.p
E L~(V), t.p
=f. 0, H(x, >.)t.p 2: "(t.p
and
T(>') = sup bE X(>.)} . It is easy to see that T(>') is nonincreasing.
\;f
xE
n}
Topics in Neutron Transport Theory
120
Theorem 5.23 r,(K(X - T)-l) 2 r(X). Moreover, if lim r(X)
A+-q
>1
Proof: The result is trivial when r(X) = 0. Assume that r(X) let y > 0, y E X(X). There exists cp E L:(V) such that /a(.,
v', X)k(v, d)cp(d)dp(d)
> 0 and
> 79 , V x E R.
Then $(x, v) = cp(v) E LP(R x V; dxdp(v)) and
whence ru(K(X - T)-')
L7
and this ends the proof since y E X(X) is arbitrary. 0 The interest of Theorem 5.23 resides in the fact that r(X) can be estimated. We illustrate this by the following examples. We assume that dp(v) = dv V={v;O
Moreover, R is assumed to be strictly convex. Then xE where
a + s(x, v) is continuous for any v # 0
n is the closure of R and consequently, for each X 2 xE
+ a(x, vl, A)
We note that s(x, 211) 2
is continuous.
dist (x, aR) Iv'I
;XER
-r]
and v'
# 0,
(5.17)
121
Chapter 5. On the leading eigenelements and, for any x E
an, there exists an open hemisphere of sn-l TI(x) = {v E sn-l; v.n(x) > o}
such that
,
I
SeX, v) > 0 ;
vTJI E II(x).
The hemisphere TI(x) is the set of unit velocities pointing outside 0. at x E an. We set sn-l if x E 0. II(x) = { TI(x) if x E an. Thus
I I
sex , v) > 0 ;
TJI E II(x) , x E 0.. V
-
(5.18)
The basic consequence of (5.18) is that 't/ x
En,
a(x ,. , A) > 0 on a subset of V of positive measure.
Corollary 5.2 Let G(v) =
-rCA) Moreover: (i) If ~£I)
~ xinf Efl E
inf vEv
k(v, v' ) > 0 a.e. Then
J
G(v')a(x, v', A)dv' = leA) > 0 ; A> -T]
Ll(V) then lim>'-+-'7 I(A) =
infxEfl
J G(v')a(x, v', -T])dv'.
(ii) If for any solid cone C with vertex zero ~£I) fj. Ll(C) , then lim leA) =
>'-+-'7
+00.
Proof: Let cp(v) = 1. Then H(x , A)cp
I ~J
=
a(x , v' , A)k(v, v')cp(v')dv'
G(v' )a(x, v' , A)dv'
~ leA) =
I(A)cp
whence -rCA) ~ leA) . It follows, from (5.17) and (5.19) , that
x
(5.19)
En
--t
lex, A) =
J
G(v' )a(x, v', A)dv'
122
Topics in Neutron Transport Theory
is continuous and strictly positive, so that l(A) is continuous and for any x En; l(x, A)
--+
> O. In the case (i), l(., -T])
l(x, -T]) as A
--+
-T].
Using a decreasing sequence Aj --+ -T] and Dini's theorem, l(x, Aj) l(x, -T]) uniformly on and consequently
n
lim l(A) = inf
xEfl
)..-+-7)
f
--+
G(v')a(x, v', -T])dv' .
In the case (ii), by the monotone convergence theorem, l(x, Aj)
--+
f
(s(x ,v ' ) ,
dv' G(v') Jo
e-(O'(v )-7)sds
= l(x , -T]).
We note that l(x, -T]);::::
where w'
=
ful.
f
,,'
dv' G(v')
PTEI1(x)
f
S(X ,V ' ) ,
e-(0'(v)-7)sds 0
On the other hand, in view of (5.16),
(O'(v ' ) - T])
Iv'l
:S c <
00 .
We note also that if x E n then s(x,w') ;::::dist(x, an) for any w' E sn-l . If x E an, we choose a solid cone C(x) C II(x)
such that s(x,w') ;:::: cx for all that
Jful ,
dv EI1 (x)
I
G( ')
1
v
0
V -1-'-1
E C(x) for a constant c x
Wi
S(X w') '
-
e
Iv'l
( a ( v')-!))'T
_ 00
dT. -
> O. It follows
Vx E
-n.
Chapter 5. O n the leading eigenelements
&
123
a. Finally l ( x ,Aj) + oo uniformly on a
B y using Dini's theorem,
+0
uniformly o n
and limx,-, l(A) = oo.0 Similar arguments yield Corollary 5.3 Let k(., v ' ) be continuous on I/ and k ( v ,v ' )
T ( A ) >_
inf
I
(x,~)ERxV
Moreover: (i) I f k(u, v ' )
> 0 a.e. Then
k ( v ,v ' ) a ( x ,v ' , X)dvi = q A ) > 0 ; A > -?
< ~ ( u ' where )
E
L 1 ( V ) then
$ L 1 ( C ) for any solid cone C with vertex zero then
(ii) If
For degenerate scattering kernels we have a more precise result
zgl
Corollary 5.4 Let k ( v , v ' ) = f j ( v ) g j ( v ' )where f j E L?(V), S,. E LII+(V)(p-l 9-I = 1 ) . Let h i j ( v f )= g i ( ~ 'f )j ( v l ) and
+
mij(A)=inf
xEn
/
Let M ( X ) be the matrix { m i q (A); j 1 Proof: Let cp E LP(V), then
Let us look for cp 2 0 o f the form
hij(d)a(x,vl,~)dv'.
< i,j
5 N ) . Then r ( A ) 2 r,(M(X)).
Topics in Neutron Tkansport Theory
124
and y such that H ( x , X)cp 2 ycp for all x E R, i.e.
It suffices that
and clearly (5.20) is satisfied if
where ,B(X) is the vector {Pi(X)). TO solve (5.21) it sufEces to take y = T,(M(X)) and ,B(X) a non-negative eigenvector of M(X) associated with the eigenvalue y. 0
Remark 5.8 More precise exzstence results can be obtained by comparison arguments (see [36] Theorems 5 and 6).
5.8
Strict monotonicity properties of the leading eigenvalue
We are concerned in this section with the monotonicity dependence of the leading eigenvalue with respect to the parameters of the transport operator, i.e. the spatial domain, the collision frequency and the collision operator. Let R C Rn be smooth open and bounded. For each collision frequency a(.,.) E LY(R x V) and non-negative collision operator
we define the corresponding transport operator on IP(R x V; dxdp(v))
where
Chapter 5. On the leading eigenelements
125
with domain independent of a( ., .)
D(Tq)
={fEV(OXV;dXdJ.L(V)); v. ~~ EV, flL =o}.
We assume, as usual, that K(>.. - Tq)-1 is power compact on LP(O x V;dxdJ.L(v)). When aas(Tq + K) =I- 0, we denote by >"(Aq,K) the leading eigenvalue of Aq,K . Theorem 5.24 Let K and K be two collision operators such that
K 5: K ; K =I- K . If aas(Tq + K) =I- 0 then aas(Tq + K) =I- 0 and >"(Aq,K) 5: >"(Aq,K) · If K(>.. - T)-1 is irreducible then >"(Aq,K) < >"(Aq,K) . Proof: Let >"1 = >"(Aq,K), then rq(K(>"1 - Tq )-I) = 1.
On the other hand,
According to the first part of (5.22),
whence there exists >"2
~
>"1 such that rq(K(>"2 - Tq )-I) = 1
so that aas(Tq + K) =I- 0 and >"2 = >"(Aq K) . If K(>" - T)-1 is irreducible then (5.22) implies, according to Theore~ 5.13,
so that
rq(K(>"1 - Tq )-1) > 1 and therefore there exists >"2 > >"1 such that
which ends the proof. <> Similarly we have
Topics in Neutron lransport Theory
126
Theorem 5.25 Let al(.,.) and a2(., .) be two collision frequencies such that al(.,. ) 2:: a2(" ' ) j al(" ') i a2(" .).
We assume, in addition, that the spectral bounds ofTu; (i = 1,2) are the same. Ifaas(Tu1+K) i 0 thenaas (Tu2 +K) i 0 andA(Au1 ,K)::; A(A u2 ,K). If K(A - T ( 2)-1 is irreducible then A(Au1,K) < A(A u2 ,K) . Proof: The condition on the spectral bound is automatically satisfied if the velocity space is bounded away from zero. If 0 E V and if, for instance, the collision frequencies are homogeneous, this condition means
On the other hand,
for A greater than the spectral bound of Tu;. The first part of (5.23) implies
which proves the first claim. If K(A - T (2 )-1 is irreducible, then using Theorem 5.13, and we conclude as previously. <> We consider now the monotonicity with respect to the spatial domain. To this end we assume that the collision frequency a(.) and the collision operator K are homogeneous. Theorem 5.26 Let f!l and f!2 be smooth open and bounded subsets of Rn such that f!l C f!2 j f!l i f!2 '
Let Tl and T2 be respectively the streaming operator in V(f!l x Vj dxdl1(v)) and in V(f!2 x Vj dxdl1(v)} . If aas(Tl +K} i 0 then aas(T2 +K) i 0 and A(TI
+ K)
::; A(T2
If (A - T 2)-1 K is irreducible then A(TI
+ K) .
+ K) < A(T2 + K) .
127
Chapter 5. On the leading eigenelements
+K
Proof. Let >'1 be the leading eigenvalue of Tl corresponding eigenfunction. We have
and let
CPI ;:::
0 be the
(5.24) where
Sl(X,V)
= inf {s > 0iX -
sv tJ. fh}
S2(X,V)
= inf {s > OiX -
sv tJ.
Let
n2 }
By assumption
SI(X,V)
~
S2(X,V) i (x,v)
It follows from (5.24) that, on
=
nl
xV.
(5.25)
n2 x V, CPI
'l/JI
E
{
on
nl
o on n2 - nl .
is less than or equal to (5.26) where Xl is the indicator function of the set n l . We define, on V(n2 x V; dxdJ.l.( v)) , the operator
According to (5.26),
Where Xl is the multiplication operator by Xl (.). Moreover
(5.27) Thus, the first part of (5.27) yields
Topics in Neutron llansport Theory
128 and there exists X 2 2 X I such that
which proves the first claim. If ( A - T2)-lK is irreducible, then according to Theorem 5.13, r,((Xl - T ~ ) - ' K )> 1 and there exists X2
> X1
such that
which proves the second claim.
0
Remark 5.9 The irreducibility assumptions for the usual models are satisfied under positivity conditions on the scattering kernel of the form (5.7) or (5.9) when the spatial domain is convex.
5.9
Domain derivative of the leading eigenvalue
By using compactness arguments it is possible to prove, for homogeneous cross-sections, that cr,,(T + K ) depends "continuously" on the spatial domain R, where the convergence of domains is defined by
Rj + R when
xj(.)+ x(.) a.e.
as j
+ oo
where xj(.)and x(.) are the indicator functions of Rj and $2 (see [37]). In particular, the leading eigenvalue depends "continuously" on the spatial domain R. We present a recent result [14] on the differentiability of the leading eigenvalue with respect to the spatial domain for a model transport operator. Let R be a smooth bounded and wnvex open subset of Rn and
We consider the transport operator
129
Chapter 5. On the leading eigenelements
where a is a constant. It is known (see, for instance, [36]) that aas(Tn) ¥- 0 at least for f2 large enough (i.e. contains a ball of radius large enough). We denote by )'(f2) the leading eigenvalue. We are concerned with the differentiability of f2 ~ )'(f2). Let us explain the meaning of this concept. Let Cl be the space of bounded continuous vector fields 8:Rn~Rn
having bounded derivatives up to order two. Let f2 be such that aas(Tn) 0. Let f2l1 = (I + 8)f2.
¥-
By continuity of the leading eigenvalue with respect to the domain
aas(Tno) when 8 lies in a neighborhood of 0 in A: 8 E
¥- 0
Cl. Thus, the mapping
cl ~ )'(f2l1) E J =
]-a, oo[
is well-defined in the neighborhood of the origin. Definition 5.2 We say that the leading e'igenvalue ). is differentiable at f2 if A is Prechet differentiable at the origin. We denote by).' (f2, 8) the derivative in the direction 8, i.e. ),'(f2,8) = A'(O)(8) where A'(O) is the Frechet derivative of A at O. Let
fn
Let gn(x)
> 0 be the leading eigenfunction normalized by
k(i
= Iv fn(x, v)dv. E(TJ,x)
=
1
fn(x,v)dv)2dx = 1.
Finally let dt n n ; TJ E J , x E R
00
e-(U+'1)tt
Ixl
-
{O} .
Theorem 5.27 Let f2 be of class C2 and let its Gauss curvature be bounded below by a positive constant. Then). is differentiable at f2 and ).' (f2, 8)
=-
IL(~) Ian g~(x)8(x).n(x)ds(x) ;
8E
where n( x) is the outward normal at x E af2 and
1L(f2)
=
Jlnxn r
E aa ()'(f2), x - y)gn(x)gn(y)dxdy. TJ
cl
Topics in Neutron Ti-ansport Theory
130
We note that P ( R )< 0 so that the sign of X'(R,6) is given by that of
Estimates of ~ ( 0are) given in the following
Theorem 5.28 Under the previous conditions,
where d is the diameter of R and w, is the area of the unit sphere of Rn Assume that n = 2 or 3 and set
then
and
Remark 5.10 The assumption concerning the curvature is only intended to ensure that Re remains convex for small 6. The proofs being rather technical we refer the reader to [14].
5.10 An approximation theory of the leading eigenelements The aim of this section is to present an approximation theory with error estimates of the leading eigenelements of transport operators. We focus our attention only on the theoretical aspects of this method and refer to [5] for its practical implementation. We consider a transport operator with non-negative and homogeneous cross-sections
where
8f dx
Tf = - v - - ~ ( v ) f ( x , v )
; f E D(T)
131
Chapter 5. On the leading eigenelements with domain D(T) =
{f
K'P =
J
and
E LP(r! x
Vj dxdv)j v . ~~ E LP, flL = O}
k(v,v')'P(x,v')dv'
j
'P E LP(r! x Vjdxdv)
where r! is convex and
v=
{v E RnjO::;
We assume that aas(T + K)
i= 0,
Cl ::;
Ivl::; C2 < oo}.
i.e.
where -liminfv ..... oa(v) if 0 E V 'T/=
{
-00
otherwise.
The leading eigenvalue).' is determined by the equality
We assume also that
(oX - T)-l K is irreducible. We describe the approximation method. The spectral problem
is converted into In order to derive error estimates, it is more convenient to deal first with the smoother unknown 'P = K 1/J which is solution of (5.28)
and then 1/J is recovered by means of
We set
Topics in Neutron Transport Theory
132
The principle of the method consists in projecting (5.28) onto a finite dimensional subspace of piecewise constant functions. To this end we introduce a partition {Al", ... , A;;',,.} of n x V and the projection p",
7rm J(x,v) = LJiXAi"(x,v) ; (x,v)
E
nx V
i=l
where
1(Am) meas i
Jim =
1 "" Ai"
J(x ,v )dx dv
1:::; i
:::;Pm'
Let
df' =
diameter(Ar')
1:::; i
:::; Pm·
We assume that d m = max: {df'; 1 :::; i :::; Pm} -+ 0 as m -+ 00
which implies that 1Tm -+
I strongly in IJ' (n x V) .
Now we replace the operator H>. by the matrix
Hm(>\) = 7rmH>.7rm · It can be proved (see [13]) that Hm(>\) is irreducible and that
lim Pm().,)
m-+oo
= p().,)
uniformly on compact subsets of j7],00[
where so that lim r C1 (Hm ().,)) > 1 for m large enough.
>'-+-1)
Finally, we define
>:m as the solution of
and
The approximate eigenelements are ('l/Jm, >:m) where
().,m
+ a(v))'l/Jm + v.
The main mathematical result is
a'l/Jm ax =
133
Chapter 5. On the leading eigenelements Theorem 5.29 We assume that 1 < p <
er(.)
00.
Let
E Wl,CO(V) ; k(.,.) E Wl,coev
x V)
with the evenness assumptions er(v)
= er( -v) ; k(v, v') = k( -v, v') = k(v, -v').
We assume that the partition {Aj", ... ,A;',,J ofn x V is regular in the sense that there exists a constant c > 0 independent of (i, m) such that meas(Ai) 2: c(di)2n ; 1 ::::: i ::::: Pm. Then there exists a constant C independent of m such that
I): - ):ml + 117/1 -7/lmIILP + Ilv. ~~ - v.
at; IILP :::::
Cdm
.
The proof of this result is rather technical and involves several mathematical ingredients. First, p('x) is shown to be convex as a consequence of the weak superconvexity of the operator [K(,X - T)-l]2 . A parametric version of results by I. Marek [31], on the approximation of positive eigenvectors of positive compact operators on Banach lattices, is also necessary. Finally, we use Wl,p Sobolev regularity of velocity averages [34]. We refer the reader to [13] for the mathematical details.
The criticality eigenvalue problem
5.11
This problem, arising in nuclear reactor applications, consists in determining the parameters of the transport operator for which the corresponding leading eigenvalue is equal to zero. The usual approach consists in embedding the problem into a one-dimensional family of problems indexed by a fictitious parameter 'Y. We recall that the kernel of the collision operator, in a fissile material, splits as
k(x, v, v' ) = ks(x, v, v')
+ kj(x , v, v')
where ks(x, v, v') describes the pure scattering while kj(x, v, v') describes the fissions. We point out that kj(x, v, v') is separable in the physical applications. The criticality problem is then decomposed into two problems. Firstly, the spectral problem
o
=
-v. ~~ - er(x, v)cp(x, v)
+-1
f (
'Y.
+
J
ks(x, v, v')cp(x, v')dl-£(v')
k j x, v, v )cp(x, V )dl-£(v') I
,
(5.29)
Topics in Neutron Transport Theory
134
where "( > 0 is the eigenvalue we look for and 'P 2: 0 is the associated eigenfunction. Secondly, we have to "adjust the composition and the geometry of the physical system" in order that "( be equal to one. We are going to give a general answer to the first problem (5.29) . It is convenient to write it abstractly as 1
Tf+Ksf+-Kf'P=O; "(>0, 'P2:0.
(5.30)
"(
It is clear that a necessary condition to solve (5.30) is that seT) < O. Thus, (5.30) is clearly equivalent to (5.31) where
KC'Y) = Ks
1
+ -Kf · "(
We restrict ourselves to the Ll setting and assume that (0 - T)-1 KC'Y) is power compact and irreducible in Ll(n x V; dxdJ-L(v)). The irreducibility is satisfied if, for instance, (0 - T)-1 Ks is irreducible. Theorem 5.30 Under the preceding assumptions, the spectral problem (5.30) has a solution if and only if
Proof: In view of Theorem 5.7, p()..) = ra((O - T)-1 KC'Y))
> O.
Moreover, by analyticity arguments, p()..) is strictly decreasing in ).. (and continuous) so that the first part of (5.32) is necessary. We observe that (0 - T)-1 KC'Y) 2: (0 - T)-1 Ks
and, if K f -=I- 0, (0 - T)-1 KC'Y) -=I- (0 - T)-1 Ks ; V"( > O. Hence, in view of Theorem 5.13,
which shows that the second part of (5.32) is also necessary because nonnegative eigenfunctions of (0 - T)-1 KC'Y) are necessarily associated to its spectral radius (a consequence of the irreducibility assumption). Finally lim (0 - T)-1 KC'Y) = (0 - T)-1 Ks
-y->oo
in the operator norm
135
Chapter 5. On the leading eigenelements and ru((O - T)-l Ks) < 1 imply that
ru((O - T)-l K(r)) < 1 for 'Y large enough because of the upper semicontinuity of the spectral radius for the norm operator topology ([25]). Hence there exists a unique 'Y > 0 such that
ru((O - T)-l K('Y)) = 1 and this ends the proof. 0 We end this section by giving practical conditions on the cross-sections implying the assumptions of Theorem 5.30. We restrict ourselves to homogeneous cross-sections and suppose that lim inf O'(v) v ...... O
=
inf O'(v)
vEV
=",.
Proposition 5.6 (i) Let d be the diameter of 0, then
[1
II::;
II(O-T)-lKs
suPJ v' EV
-u(v)..
(5 .33)
0' V
(ii) If ru((O - T)-l Kf) > 0 then lim ru((O - T)-l K(r))
1' ...... 0
= 00.
Proof The second part follows trivially from the inequality 1
ru((O - T)-l K(r)) ~ -ru((O - T)-l Kf). 'Y
We consider (i),
(O-T)-lKscp= so that
l
S (X'V)
e-u(v)sds
J
k s (v,v')cp(x-sv,v')dJ.1,(v')
J
1(0 - T)-lKscp(x, v)1 dx
n ..
<
J
e-u(v)sds
o
J
k s(v,v')dJ.1,(v')
J
Icp(y,v')1 dy
(5.34)
n
where d is the diameter of O. Integrating (5.34) over V yields the upper estimate (5.33) . 0
136
Topics in Neutron Damport Theory
Remark 5-11 If d p ( v ) = dv, k f ( v ,v ' ) = x i ( v ) x a ( v l ) and^(.) = X I ( . ) X Z ( . ) # 0 is even (or dominates an even function) then r , ( ( 0 - T)-I Kf) > 0 (see [361).
5.12 The effects of delayed neutrons This section is devoted to the peripheral spectral theory of transport o p erators with delayed neutrons. We restrict ourselves to wntinuous models. The analysis of multigroup models is similar. We consider the system
and assume that k(. , ., .) ( 0 5 i _< m) are non-negative. It is clear that the semigroup { V ( t ) ;t 2 0) governing this system is positive since the unperturbed semigroup { U ( t ) ;t L 0 ) and the perturbation B are positive. We refer to Section 4.5 for the different notations and assumptions. We assume that
A1
; -A1>r]
and that the velocity space V contains a spherical shell
We first show that a leading eigenvalue always exists under very general assumptions
Theorem 5.31 Let there exist a ball Ro c R such that
then A has a real eigenvalue
> -A1.
Proof: We consider the spectral problem
137
Chapter 5. On the leading eigenelements
where Q = (fo, f i , ...,f m ) l . It is easy to see that (5.37) is equivalent to
where
According to Theorem 4.1 or Theorem 4.4, (X-T~)-'K(A) is power compact (and positive). Hence the leading eigenvalue exists if and only if
lim r, ((A - TO)-'K(x)) > 1
A--A1
and the leading eigenvalue 5; is characterized by the equality
Note that - To)-'K(X))
L
A ~ ~ ~- TO)-'KI) ( ( A
A+A1
Thus it suEces that T , ( ( A - T ~ ) - ~ K ~>) 0. It is not difficult to see that the spectral radius of (A - To)-lKl as operator on P ( R x V) is greater than or equal t o its spectral radius as operator on P ( R o x V). The assumption of the theorem ensures the irreducibility of (A - To)-'K1 on P ( R o x V) (its square is positivity improving) so we conclude by Theorem 5.7. 0 We end this section by giving an irreducibility criterion of the semigroup {V(t); t L 0).
Theorem 5.32 We assume that
and, for any i E [1, ...,m ] , there exists K sure such that
C
V with positive Lebesgue mea-
then {V(t); t 2 0) is positivity improving (and hence irreducible) for t where p-diam(R) is the path diameter of Q. c2
>
Topics in Neutron llansport Theory
138
Proof: Let {Vo(t);t 2 0 ) be the semigroup on P ( R x V ) generated by To KO. According to Theorem 5.14, Assumption (5.39) ensures that { b ( t )t; 0 ) is positivity improving for t > f = . We note that system (5.35), (5.36) may be written as
+
>
i
f i ( x , v , t ) = e - X i t f i ( ~ ,+ ~ ) e - X i ( t - S ) ~ i f O ( ~ ;) d( 1~
o
+ EmXi a=1
I"
0
KJ (t - s )
Is
e-k(s-T) K j fo ( r ) d r .
0
Hence, for non-negative initial data and t
f ( v , t ) 1 Ki
>f,
fo(s)ds ; ( 1 5 i
Let Q0 = ( f O , f l , ...,f m ) # 0. By noting that each ( 1 i 5 m) is positivity improving, one sees that
<
< i 5 m)
I$
< m).
(5.41)
e-x'(t-s)~
which implies, by the second inequality and (5.40), that
and the proof is complete. 0 The following shows that Assumption (5.40) is natural.
Theorem 5.33 Let there exists i E [ I ,...,m] such that
where Woc R x V has a positive Lebesgue measure. Then { V ( t ) ;t reducible.
2 0 ) is
139
Chapter 5. On the leading eigenelements
Proof. We may suppose that n x V - Wo has a positive measure. Let fi ~ 0, f i =J 0, fi(x, v) = on Wo and i 1 o fl ,T,O 'J.' = (f " , ." f - , fi '" 0' fm) .
°
Then \lIo =J 0. Let ~ E [Lq(n x V)]m+1 (p-l + q-l = 1) whose components are equal to zero except the i-th one which is non-negative and equal to zero on n x V - Wo o Then (V(t)\lIo,~) = thus showing the reducibility of {Vet); t ~ o} in view of Proposition 5.4 (ii). 0
°
5.13
Comments
As we pointed out in the introduction, and showed in the previous sections, positivity plays a crucial role in the description of the peripheral spectrum of transport operators. Its importance was emphasized very early, particularly by G. Birkhoff [7] [8] [9], in the context of criticality. The analysis of the boundary spectrum of transport operators started with I. Vidav's paper [45] and was continued, in particular, by J. Mika [32], T. Hiraoka and S. Uka} [22] [23], N. Angelescu and V. Protopopescu [3], E.W. Larsen [28], I. Marek [29], G. Greiner [18] [19], J. Voigt [46] and M. Mokhtar-Kharroubi [35]. Positivity also enters into play in the analysis of compactness problems in transport theory through dominations theorems by P.G. Doods and D.H. Fremlin [17] and C.D. Aliprantis and O. Burkinshaw [1] as was shown in Chapter 4. As we noted in Section 5.1, positive semigroups and their spectral theory are extensively presented in R. Nagel et al [38] . The general theory of positive operators and their spectral theory are covered by H.H. Schaefer's treatise [41] . We recommend particularly the refreshing survey by M. Zerner [49] which provides a nice introduction, for applied mathematicians, to spectral theory of positive operators. We also mention the book by A. Berman and R.J. Plemmons [6] where spectral results on nonnegative matrices are extensively presented. The material in Section 5.10 and Theorem 5.11 are taken from V. Caselles and M. Mokhtar-Kharroubi [13] while Theorem 5.12 is due to the author (unpublished). The material in Section 5.8, Theorems 5.15 and 5.16 are taken from M. Mokhtar-Kharroubi [35]. The results in Section 5.4 are due to the author (unpublished). The use of irreducibility arguments and weak compactness of transport semigroups (for large times) in Ll spaces, to prove the existence of eigenvalues by means of H.H. Scheafer's result [42], was first mentioned by J. Voigt [46]. On the other hand, Theorem 5.22 asserts that, if the leading eigenvalue >. exists, then it satisfies the estimate d
foe e-(A+infa(,))sds x raCK) ~ 1.
140
Topics in Neutron Transport Theory
This suggests a link between the spectrwn of the transport operator on
V(O x V) and that of the collision operator K as operator on V(V). This link will be emphasized further in Chapter 6. The material in Sections 5.5, 5.6 and 5.7 is taken from M. Mokhtar-Kharroubi [35] [36] . The results in Section 5.9 are due to M. Choulli, A. Henrot and M. Mokhtar-Kharroubi [14]. Finally, Sections 5.11 and 5.12 are an expanded version of M. MokhtarKharroubi [33] Chapter VI. An approximation theory for the eigenelements associated to the spectral radius of irreducible operators is given by I. Marek [31] and adapted to transport operators by V. Caselles and M. MokhtarKharroubi [13] (see also Section 5.10). The practical implementation of this approximation method is given by F. Arandiga and V. Caselles [5] (see also F. Arandiga [4]). For eigenvalue calculations we refer to B. Dahl [16] and the references therein. The strict monotonicity results of Section 5.8 are based on strict comparison (of spectral radius) results by I. Marek [30] . For strict monotonicity results for critical eigenvalues and related results we refer to G. Borgioli, G. Frosali, C. van der Mee [10] and the references therein (see also M. Jung [24]). A fine analysis of leading eigenelements for integral operators (motivated by transport theory) is given in H.D. Victory [44] (see also P. Nelson [39]). The equality of the spectral bound and the type for general positive semigroups has been known for a long time in L1 spaces or Hilbert spaces and the problem of whether it extends to V spaces (1 < p < 00) was open until recently where it was solved in the positive by L. Weis [48] . We point out that prior to L. Weis's result, J. Voigt [47] proved, by interpolation argwnents, a convexity property of the spectral bound for positive semigroups having the interpolation property in V spaces, from which he deduced the identity of the spectral bound and the type for such semigroups. L. Weis's result [48] is used in transport theory by F. Ammar-Khodja and M. Mokhtar-Kharroubi [2] to estimate the type of streaming semigroups in the presence of boundary operators. The irreducibility criteria for transport semigroups given in Section 5.3 are only concerned with the usual continuous and multigroup models, thus covering the practical situations. However, from a mathematical view point, it would be interesting to study the irreducibility of transport semigroups for a general velocity measure dl1-( v) in order to understand the role (if any) of such a measure. In contrast to the case where the velocity space is bounded away from zero, where we have at our disposal general criteria of existence of eigenvalues (see Section 5.4), a general criterion of existence of eigenvalues when 0 E V is still lacking in spite of the various existence results of Section 5.7 and [22] [23] [36].
Chapter 5. On the leading eigenelernents
141
References [l]D. Aliprantis and 0 . Burkinshaw. Positive compact operators on
Banach lattices. Math. 2. 174 (1980) 289-298. [2] F. Amrnar-Khodja and M. Mokhtar-Kharroubi. On the exponential stability of advection semigroups with boundary operators. To appear in M3AS: Math. Models Methods Appl. Sci. [3] N. Angelescu and V. Protopopescu. On a problem in linear transport theory. Rev. Roum. Phys. 22 (1977) 1055-1061. [4] F. Arandiga. Aproximacion de operadores y continuidad del radio espectral. Thesis, 1992, Universitat de Valencia. [5] F. Arandiga and V. Caselles. Multiresolution approach to the approximation of the leading eigenelements of some neutron transport operators. Damp. Theory Stat. Phys. 25(2) (1996) 121-149. [6] A. Berman and R.J. Plemmons. Nonnegative matrices i n the mathematical sciences. Academic Press, 1979. [7] G. Birkhoff. Positivity and criticality. Proc. Symp. Appl. Math. XI. Amer. Math. Soc. Providence. R.I. 1961. [8] G. Birkhoff. Reactor criticality in transport theory. Proc. Nut. Acad. Sci. 45 (1959) 567-569. [9] G. Birkhoff. Reactor criticality in neutron transport theory. Rend. di. Math. 22 (1963)102-126.
[lo] G. Borgioli, G. Frosali and C. van der Mee.
Comparison of the critical eigenvalues for integral neutron transport equations in different geometries. Dansp. Theory Stat. Phys. 14(2) (1985) 223-252.
[ll]V. Caselles. On the peripheral spectrum of positive operators. Israel J. Math. 58(2) (1987) 144160.
[12] V. Caselles. An extension of Ando-Krieger's theorem to ordered Banach spaces. Proc. Amer. Math. Soc. 108(4) (1988) 1070-1072. [I31 V. Caselles and M. Mokhtar-Kharroubi. On the approximation of the leading eigenelements for a class of transport operators. D a m p . Theory. Stat. Phys. 23(4) (1994) 501-516.
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1141 M. Choulli, A. Henrot and M. Mokhtar-Kharroubi. Domain derivative of the leading eigenvalue of a model transport operator. Submitted. 1151 Ph. Clement et al. One-Parameter Semigroups. North-Holland, 1987. [16] B. Dahl. Numerical solutions of the monoenergetic neutron transport equation with anisotropic scattering. Thesis, 1985, Chalmers University of Technology, Sweden. [17] P. Doods and D.H. Fremlin. Compact operators in Banach lattices. Israel J. Math. 34 (1979) 287-320. [18] G. Greiner. Spectral properties and asymptotic behavior of the linear transport equation. Math. 2. 185 (1984) 167-177. [I91 G. Greiner. An irreducibility criterion for the linear transport equ& tion. Semesterberircht Functionalanalysis, Tubingen, (1984). 1201 G. Greiner. Asyrnptotics in linear transport theory. Semesterberircht Functionalanalysis, Tubingen, (1982). 1211 G.J. Habetler and M.A. Martino. Existence theorems and spectral theory for the multigroup diffusion model. Proc. Symp. Appl. Math. XI. Amer. Math. Soc. Providence. R.I. 1961. 1221 T. Hiraoka and S. ukal. Eigenvalue spectrum of the neutron transport operator for a fast multiplying system. J. Nucl. Sci. Tech. 9(1) (1972) 36-46. [23] T. Hiraoka and S. ~ k a i .Conditions of fast multiplying systems for existence of the time eigenvalue of the fundamental mode. J. Nucl. Sci. Tech. lO(3) (1973) 54-59. 1241 M. Jung. Classification of the integral neutron transport operator and analysis of its spectral radius. Dansp. Theory Stat. Phys. 22(6) (1993) 861-869. [25] T. Kato. Perturbation Theory for Linear Operators. Springer Verlag, 1984. [26] T. Kato. Superconvexity of the spectral radius, and convexity of the spectral bound and the type. Math. 2. 180 (1982) 265-273. [27] J.F. C. Kingman. A convexity property of positive matrices. Quart. J. Math. Oxford. 12(2) (1961) 283-284.
Chapter 5. On the leading eigenelements
143
[28] E.W. Larsen. The spectrum of the multigroup transport operator for bounded spatial domains. J. Math. Phys. 20(8) (1979) 1776-1782. [29] I. Marek. Fundamental decay mode and asymptotic behaviour of positive semigroups. Czech. Math. J. 30(105) (1980) 579-590. [30] I. Marek. Frobenious theory of positive operators: Comparison t h e e rerns and applications. SIAM J. Appl. Math. 19(3) (1970) 607-628. [31] I. Marek. Approximation of the principal eigenelements in K-positive non-self-adjoint eigenvalue problem. Math. System. Theory 5 (1971) 204215. [32] J. Mika. Fundamental eigenvalues of the linear transport equation. J. Quant. Spectr. Radiat. Transfer. 11 (1971) 879-891. [33] M. Mokhtar-Kharroubi. Les equations de la neutronique. Thtse d'Etat, 1987, Paris. [34] M. Mokhtar-Kharroubi. WIJ' regularity in transport theory. Math. Models Methods Appl. Sci. l ( 4 ) (1991) 477-499. [35] M. Mokhtar-Kharroubi.Quelques applications de la positivitk en theorie du transport. Ann. Fac Sci. Toulouse. X I (1) (1990) 75-99. [36] M. Mokhtar-Kharroubi. Some spectral properties of the neutron transport operator in bounded geometries. Tbansp. Theory Stat. Phys. 16(7) (1987) 935-958. [37] M. Mokhtar-Kharroubi. C.R. Acad. Sci. Paris. Ser I. 297 (1983) 331-334. [38] R. Nagel et al. One-parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, 1184, Springer-Verlag, 1986. [39] P. Nelson. The structure of a positive linear integral operator. J. Lond. Math. Soc. 8(2) 1974) 711-718. [40] B.de. Pagter. Irreducible compact operators. Math. 2. 192 (1986) 149-153. [41] H.H. Schaefer. Banach Lattices and Positive Operators. Springer, 1974. [42] H.H. Schaefer. On the spectral bound of irreducible semigroups. Semesterberircht Functionalanalysis. Tubingen, (1983).
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[43] S. Steinberg. Meromorphic families of compact operators. Arch. Rational Meeh. Anal. 31 (1969) 372-379. [44] H.D. Victory. On linear integral operators with nonnegative kernels. J. Math. Anal. Appl. 89(2) (1982) 420-441. [45] I. Vidav. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22 (1968) 144-155. [46] J . Voigt. Positivity in time dependent transport theory. Acta. Appl. Math. 2 (1984) 311-331. [47] J . Voigt. Interpolation for (positive) eo-semigroups on Lp-spaces. Math. Z. 188 (1985) 283-286. [48] L. Weis. The stability of positive semigroups on Lp-spaces. Proc. Amer. Math. Soc. 123(10) (1995) 3089-3094. [49] M. Zerner. Quelques proprietes spectrales des operateurs positifs. J. Funct. Anal. 72 (1987) 381-417.
Chapter 6
Spectral theory of transport operators with form positive collision operators 6.1
Introduction
The general aspects of the asymptotic spectrum of transport operators, tied to compactness results, have been described in Chapter 4 while the peripheral spectrum of transport operators, related to positivity, was dealt with in Chapter 5. We investigate in this chapter a class of transport operators for which a much finer description of the real point spectrum is possible. This class rests on the possibility of symmetrizing the eigenvalue problem for transport operators when the collision operator is positive for the inner product. The theory is completely hilbertian and does not rely on any positivity assumption in the lattice sense. Let n be a smooth and bounded open subset of R n and let dfJ- be a positive Radon measure on R n invariant by symmetry with respect to the origin. We denote by V the support of dfJ- and assume that the hyperplanes'oJ Rn have zero dfJ--measure. This covers, of course, the usual continuous or multigroup models. We are interested in the eigenvalue problem for the operator
8cp Acp = -v. 8x - a{v)cp(x, v) + Kcp = Tcp + Kcp ; cp E D(A)
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Topics in Neutron ?lamport
146
Theory
with domain
f
E L2(R x
V ) ; v.-af ax
E
L2(R x V ) , fir- = 0
where
L'(R x V )= L'(R x V ; d ~ d p ( ~ ) ) and
-'I
= { ( x ,V ) E 8R x V ; v.n(x) < 0 ) ,
n ( x ) being the outward normal at x E 8R. The assumptions on the collision frequency are a(.) E Lm(V;dp) , a ( v ) = a(-v) (6.1) while the collision operator
K : cp E L2(V;dp) + Kcp =
J
k(v,d)cp(vt)dp(v')E L 2 ( v ;dp)
is assumed to satisfy the following conditions
K : L2(V;dp) + L'(v; dp) is compact (6.2) k(v,v') = k(-v, v t ) = k(v, -vl) and
K : L'(v; dp) + L2(V;dp) is positive self-adjoint.
(6-3) The positivity of K is understood in the inner product sense. We recall that the spectral bound of T is
In the case where 0 E V , we will assume that
In such a case s ( T ) = -a(O). We consider now the spectral problem
We recall also that
Chapter 6. Form positive collision operators so that (6.5) is equivalent to
In view of (6.3) , K admits a self-adjoint positive (in L2 sense) square root Thus, the spectral problem (6.5) is equivalent to
a.
We define the basic operator
In section 6.2, we show that HA is self-adjoint, compact and derive an expression for the corresponding quadratic form by Fourier transform. We also give some spectral properties of the bounded part of the transport o p erator. In view of the self-adjointness of HA,the real point spectrum of the transport operator is completely characterized in terms of the curve eigenvalues of HA.Thus, the spectral theory of the transport operator reduces to estimating the curve eigenvalues of HA.This is achieved by looking for suitable test functions. This program is carried out in Section 6.3 for given spatial domains, where very precise existence criteria are given as well as estimates of the number of eigenvalues. In section 6.4, we study the behaviour of the point spectrum as the spatial domain gets large and show that the number of eigenvalues increases indefinitely and all the eigenvalues tend to the leading eigenvalue of the bounded part of the transport operator. In section 6.5, we deal with isotropic models. We show a reality result and give a necessary and suficient criterion of finiteness.
6.2
Self-adjointness and quadratic form
We begin with basic properties of HA
Theorem 6.1 HA is a self-adjoint compact operator on L~( a x V; dxdp(v)).
a
Proof: Since is compact in L ~ ( vdp) ; (its square is compact and is self-adjoint), then *(A - T)-' is compact in L2(R x V) in view of
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Theorem 4.1. On the other hand,
(H). 7/J, cp)
= (.JK(,X =
=
T)-l.JK7/J, cp)
= «,X -
j nxv
.JKcp(X, v)dxdf-l(v)
j nxv
.JKcp(x, -v)dxdf-l(v)
j
T)-l.JK 7/J, .JKcp)
S(X 'V)
e-(A+t7(v»s.JK7/J(x - sv, v)ds 0
j
S(x.-V)
e-().+t7(-v))s.JK7/J(x + sv, -v)ds.
0
From the spectral representation of Kin L2(V; df-l) and the evenness of its eigenfunctions (a consequence of the evenness assumptions on k(v, Vi)), it follows that the range of.JK is composed of even (with respect to velocities) functions. Thus, the evenness of 0-(.) shows that
= =
I
. nxv
.JKcp(x, v)dxdf-l(v)
j
S(x .-V)
e-().+t7(v»s.JK7/J(x + sv , v)ds
0
j nxv .JKcp(x, v)(,X - T*)-l.JK7/J(x, v)dxdf-l(v)
which shows the self-adjointness of H). . <> For ,X > - inf 0-(. ), we have the further information Theorem 6.2 Let and
n be
where ~(w, v) = (27r)-n/2
convex and ,X > - inf a(.) . Then H). is positive
In 7/J(x , v)e-iX'Wdx .
Proof. In view of the convexity of n,
149
Chapter 6. Form positive collision operators where E
n,
is the restriction operator, and
Hence
(H)..'l/J, 'l/J)
= (../K(>. - T)-l../K'l/J, 'l/J)P(OXV)
= (../KRo(>' - Too)-l E../K'l/J, 'l/J)P(OXV) = (Ro../K(>. - Too)-l../KE'l/J, 'l/J)P(oxV) = (Ro../K(>. - Too)-l../KE'l/J, E'l/J)L2(OXV) = (../K(>. - Too)-l../KE'l/J, E'l/J)P(RnxV)
= ((>. - Too)-l../KE'l/J, ../KE'l/J)P(RnXV) '
On the other hand, the Fourier transform~(with respect to the space variable) of (>. - Too)-l¢ is equal to 1
~
>.+O'(v) +iv.w¢(w,v) where w is the dual variable. Parseval's identity yields
IJKE'l/J(w,v)12
J J+ dJ-t(v)
v
Rn
>.
O'(v)
+ iv.w
dw
Finally, by decomposing the integrand into real and imaginary parts, one sees that the latter is odd with respect to velocities because ~(w, .) is even. Thus, the integral reduces to its real part and (6.9) follows. <> We observe at once, in view of (6.8) and the self-adjointness of H).., that the real point spectrum of T + K can be characterized as follows: Let
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PI (A) ;::: P2(A) ... ;::: Pm(A) ;::: ... be the non-negative eigenvalues of the com-
pact self-adjoint operator H).. (A > s(T)), each eigenvalue being repeated according to its multiplicity. In the case where H).. has only finitely many positive eigenvalues, the sequence is continued with the zero eigenvalue of infinite multiplicity. Then, the real eigenvalues of T + K are the solutions of the equations Pi(A) = 1 (i = 1,2, ... ). Thus, the key point is to find suitable estimates of the curves A - t Pi(A) (i = 1,2, ...). For the sequel, we need other preliminary results. As in Section 5.5, the (self-adjoint) bounded part
is expected to play a role thereafter. Its spectral structure is easy to describe. Indeed, it follows from the compactness of K that
a(B) n {A; A> -inf a(.)} consists of, at most, isolated eigenvalues with finite multiplicities. Moreover, it is easy to see that the spectral problem
Bcp = ACP, A> -inf a( .) is equivalent to
S)..7jJ:=
r JA +k(v,v')7jJ(v') dp,(v') = 7jJ(v) a(v)JA + a(v')
iv
where 7jJ(v) = JA + a(v)cp(v). Clearly, S).. is positive in the scalar product sense and compact in L2(V; dp,) so that liS).. II is its greatest eigenvalue. We note easily the continuity of
A -t
IIS)..II .
Let us show that it decreases (strictly) . A priori, this is not obvious since the kernel k(v, v') is not assumed to be non-negative. We observe however that where
M).. : cp E L2(V; dp,)
1
-t
cpo
JA+ a(.) Hence
liS).. II = IIMnlK'112 and consequently liS).. II =
sup 11'1'11=1
J+
12
A 1 ( ) lv'Kcp(v) dp,(v) a v
151
Chapter 6. Form positive collision operators
from which follows the decreasing property. Finally, since S), depends analytically on >. and IIS)'II is an eigenvalue of S), it follows, from GohbergShmulyan's theorem, that liS), II decreases strictly in >.. From these remarks we deduce easily Lemma 6.1 a(B) n {>.; >. > - inf a(.)}
lim
),-+-
inf
In this case, the unique>: defined by
i= 0 if and
liS), II
only if
> 1.
IISAII = 1 is the leading eigenvalue of
B. We need one more preliminary result. To this end we define the following operators
..1....,jK:
v:
cp ---t
{ D( },jK)
=
v'Kcp(v)
..;v
{cp E L2(V);
v'K},}V) E L2(V)} .
We note that },jK is closed.
,jK.}.: cp { D(,jK .}.)
---t
,jK(~)
=
{cp E L2(V); ~
E
L2(V)} .
We note also that ,jK.}. is (at least) densely defined and that .}.,jK is the adjoint of ,jK .}. . Finally we set
with domain
We observe that
K
cp
= ( k(v, v')cp(v') d (v') .
The following result is clear
lv MM
J1,
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152
Lemma 6.2 The following assertions are equivalent. (i) L a is a bounded operator
J
(zi)
J is a bounded operator
(iiz) R is a bounded operator.
6.3
Existence results for given spatial domains
We first consider the case where 0 E V and assume that (6.4) holds
Theorem 6.3 a(T + K ) n {A;X > -a(O)) if and only i f I? is not a bounded operator.
# 0 regardless of the size of R
Proof: We observe that
1
s(x,v)
( A - ~ ) - l c p ( V~) , =
e-(*+u(v))scp(~ - SV, V
) ~ S
0 s(x,v)
-
L IvI /
1
- s w , v ) d s := -0~cp
e-?cp(z
lvl
(I
where w =
'E Sn-l.It is easy to see that, for all X > -c~(0), IvI
u v - U O
I~OA 5~dI x
sup { e -
'1-1
(
(
) s ;v
E V,s E [o,s ( x , w ) ] } = d x
a
where d is the diameter of R. We note that a -
1when ~ ( 0=) inf a ( . ) . Let K be bounded. Then, according to Lemma 6.2, L a and J?7$ are also J bounded and
This implies that
11q1
"
> -g(O) and consequently (6.8) has no solution if Ed llXll < 1. We point out that the IlHxll 5
1
same argument shows also that no complex eigenvalue exists. Conversely, suppose that I? is not bounded. Then, according to Lemma 6.2, the domain of the closed operator L a is not the whole space L 2 ( v ;d p ) and J consequently there exists cp E L 2 ( V ;d p ) with unit norm such that
J;;
$ L2 (v; dp)
153
Chapter 6. Form positive collision operators We define 'Ij; E £2(0 and
X
V) by 'Ij;(x, v)
= 101-t cp(v). Then 11'Ij;lbcnxv) = 1
is given by
(6.11) or to
Let PI (A) := sup (H>.J, f).
11/11=1
Thus, a lower bound of PleA) is given by
(6.12)
We point out that PleA) E a(H>.) because H>. is self-adjoint (see, for instance, [3] p. 96). We point out also that PleA) is not equal, a priori, to the norm of H>. since the latter may have, a priori, a negative spectrum if o is not convex or (in the case where 0 tJ. V) if A < - inf a(.). The estimate (6 .12) shows that PleA) > 0 and, consequently, PleA) is the greatest eigenvalue of the compact operator H>. . Thus
=
101- 1
~c
f
f f [f df-L(v)
n dx
S(x,w) 0
df-L(v) 1vfviVKcp(v) 12 =
e-
+00
,,(~) -,,(O) ) 1 12 I~I 8ds vfviJKcp(v)
Topics in Neutron Transport Theory
154
in view of (6.10) where
c = Inl
-. 1 [l I
mf
vEV
n
dx
S
(x,W)
e
-
,,(v)-,,(O) 8
Ivl
I
ds > O.
0
Since PI(>') depends continuously on >. > -0'(0) (see [2J Theorem 8, p. 213) and tends to zero at infinity, then there exists>. > -0'(0) such that PI (>.) = 1 and consequently (6.8) has a solution regardless of the size of n. This ends the proof. 0 We have seen, when K is bounded, that O'(T + K) n {>.; >. > -O'(O)} = 0 for n small enough. The following theorem gives more precise informations for constant collision frequencies. To this end we define some useful parameters. We note that dist(x, an)
Ivl
::; s
( )< x,v -
d
T0'
(6.13)
Let
s(v) :=
1~ll s(x, v)dx , 'Y:= 1~ll dist(x, an)dx.
It follows from (6.13) that 'Y
d
T0 ::; s(v) ::; T0 ' Let
K be the integral operator on L2(V; dJ..L) k(v,v')
(6.14)
with kernel
= Js(v)k(v,v')Js(v') .
It is clear, from (6.14), that
(6.15)
We have Theorem 6.4 Let K be bounded in L2(V; dJ..L) and let 0'( .) be constant. Then (i) O'(T + K) n {>.; >. > ->'*} = 0 if d <1
IIKII
(ii) O'(T +K)n {>.; >. > ->. *} = 0 if
IIKII > 1, in particular if'Y IIKII > 1.
155
Chapter 6. Form positive collision operators
Proof. (i) is a part of Theorem 6.3. Let cp E L2(V; dp,) with unit norm. Then the calculation in (6.11) shows that
=
101- 1
f f n dx
v dp,(v)
[f
S(x>V) 0
1
e-(C1+>')sds IVKcp(v)
I
2
from which we get
Hence, since cp E L2(V; dp,) is arbitrary with norm 1, taking the supremum yields lim >'~~C1P1('\) and we conclude as previously.
~ IIKII > 1
<>
Remark 6.1 This theorem shows that the existence (or nonexistence) of eigenvalues is strongly related to the size of the geometrical parameters d and 'Y.
The following example gives more insight into the (probable) physical interpretation of the existence of eigenvalues Corollary 6.1 Let V be bounded and let k(v;v') = c be a constant. Then
O'(T + K) n {A;'\ > -O'} =I- 0 if
r
inxv
s(x,v)dxdp,(v)
> ~. c
Proof. Let cp E L2(V; dp,) be arbitrary with norm 1. Then
Thus
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156
so that
11K I = sUPII~II=lIIK\p1I = c
[Iv
= c
[1 v
s(v)dtt(v) 1= Inl
and we conclude by Theorem 6.4.
.i 2
s(V)dtt (v)1
I VsDIIL2
loxv s(x, v)dxdtt(v)
0
Remark 6.2 One sees that the existence of eigenvalues has to do with the magnitude of the parameter
r
ioxv
s(x, v)dxdtt(v)
which can be viewed as the "mean time", for the particles, to reach the boundary. Remark 6.3 If K is bounded then H).. has a limit, in the operator norm, as A ---+ -0"(0) H-a(O) :=
[JK ~1
O-u(O)
[~ JKj.
It follows that only .finitely many equations Pi(A) = 1 (i = 1,2 ...) may have solutions, since Pi(A) ---+ 0 as i ---+ 00 uniformly in A E [-0"(0),00[. It is to be expected that the curves Pi(')' which contribute to the spectrum ofT+K, give rise to only finitely many solutions. (This is true, of course, if the curves are decreasing.) In such a case, the point spectrum of T + K is .finite. This is certainly true if 1 is not an eigenvalue of H -u(O) ' Anyway the number of eigenvalues of H-u(o) exceeding 1 provides a lower bound of the number of eigenvalues of T + K. The point spectrum of T + K can be infinite as is shown in the following theorem. To this end we recall that the Radon transform of \P E Ll(V; dv) is given by R\p(w, s):=
l .v=s
\p(v)dv ; wE
Theorem 6.5 Let dtt(v) = dv and O"(v) exists q E L2(V) such that
= 0"
sn-\
s E R.
a constant. We assume there
lim R(IKqI2)(w, s) = +00.
8--+0
Then the point spectrum of O"(T + K) n {A > -O"} is infinite.
(6.16)
157
Chapter 6. Form positive collision operators
Proof. Let mEN be an arbitrary integer and Fm C L2(0) be a subspace of dimension m. Let cp E L2(V) with unit norm. We define also the mdimensional subspace of L2(0 x V)
According to the max-min principle (see [2] Theorem 5, p. 212),
where
1>. E Sm.
We note that
f>,(x, v) = 'l/J)..(x)cp(v) where
According to (6.9),
(H)..J>-.,
_ r dw 1'l/J)..(w) - 12 r (a(v .w)2 + >.) IJKcp(v) 12 + (a + >.)2 dv.
j)")L2(O XV) -
iRn
iv
We set h = IJKcp(v) 12 E £l(Rn) by extending it trivially outside V. Then
Pm(>')
~
/d
1;;;( w )..
2 )1 jd
W
.
Rn
R
8
/
(a+>.)h(v)
Iwl282+ (a + >.)2
v'I::;I=s
dw 1'l/J).. (w) 12 j d8-(a--;:.-->.)_R_h....!.(1:=1_'8_) .R Iwl 82+ (a + >.)2 Rn /
=
/
Rn
N ow we ch oose cp =
1
dw 'l/J)..(w)
VKql IIVKq I'
Th
12/ d8 Rh(I:I,(a+>.)p) 2 Iwl p2 + 1 R
en
h() IKq(v)12 v = IIVKql11 and
.
d V
Topics in Neutron Transport Theory
158
Letting Aj --t -a, the sequence {'l/J>.j; j E N} , included in the unit sphere of the finite dimensional space Fm, is relatively compact. We may assume that 'l/J>.j --t 'l/J in Fm (11'l/JIIL2(f2) = 1). Since ;;;;; --t
;j in L2(Rn)
then, in view of (6.16), Fatou's lemma yields
Hence, in view of the continuity of the curves Pi(.) ([2] Theorem 8, p. 213), there exist {31> {32, .. . , (3m such that P1.·({3t·) = 1
,
1_ < i_ <m
and this shows the existence of m eigenvalues {{3i; 1 :S i :S m} of T + K in view of (6.8). This ends the proof since m is arbitrary. <> Let us consider the case where 0 ~ V. Then the spectral bound of T
seT) = We recall that K : L2(V; dJ.L) with
--t
-00.
L2(V; dJ.L) is positive compact. We begin
Theorem 6.6 Let 1 :S M :S 00 be the number of positive eigenvalues of K. Then aCT + K) n R contains, at least, M eigenvalues. Proof Let E be the orthogonal (in L2(V)) subspace to the kernel of K. It is spanned by the eigenfunctions of K corresponding to the positive eigenvalues. Let m :S M (m finite), and Em C E be a subspace of dimension m. Let C c n be a ball. We define Oifx~C
'l/J(x) :=
{
ICI-~ if x
E C
where 101 is the Lebesgue measure of C. We define also the m-dimensional subspace of L2(n x V) 8 m :='l/J®Em. Let PI (A) 2: P2 (A) 2: .. . be the non-negative eigenvalues of the compact self-adjoint operator H>., each eigenvalue is repeated according to its multiplicity. According to the max-min principle (see [2] Theorem 5, p. 212),
159
Chapter 6. Form positive collision operators where f>.(x, v)
= 1P(x)u,X(v) ; U,X( .) E Em , Ilu,Xlb(V) = 1.
Hence
J [8(7)
> IOI-!
CxV
J
101- 1
e-('x+U(V))S1P(X - SV)dS] 1v'Ku,X (v) 12
0
[SjV) e-(,X+U(V))SdS]
CxV
lv'Ku,X(v) 12 > 0
(6.18)
0
where s(x, v) = inf {s > 0; x - sv rt. C} (x E C). By choosing a sequence Aj ---7 -00, {u,Xj; j E N} belongs to the unit sphere of the finite dimensional space Em. We may assume that u'xj
---7
U in Em ,
Ilull =
1.
Hence /Ku'xj ---7 /Ku in L2(V) and /Ku =I- 0 since u E Em C E. Finally, by Fatou's Lemma, lim inf Pm(A) = +00 >"j--+-OO
and we conclude as previously. 0 We end this section by showing a link between eigenvalues of T those of K .
+K
and
Theorem 6.7 Let Q1 2: Q2 2: ... 2: Qi·· · (i :s; M) be the eigenvalues of K. Let C c be a ball with radius r and let 0 < T < 1. We assume that V is bounded. Then any solution!3i of the equation Pi(!3) = 1 satisfies the estimate (l-T)r
n
T- n
r
Jo
6
e-({3,+u)8ds:S;
~ (1:S; i :s; M) Qi
where b is the maximum speed and ~ = max a(.) . Proof We use the arguments used in Theorem 6.6. We turn to (6.18) and choose, as m-dimensional subspace Em C E, the space spanned by the first m eigenfunctions of K . We have
Topics in Neutron llansport Theory
160 where Then
,(A)
UA
E Em. Let
2 CI-'
Zi be a ball concentric with C
/_cxv dxdp(v)
[$(x'v)
and with radius ~ r .
I
1
e-(A+u(v))sds~
1. 2
U( v ) A
It follows from
that
and this ends the proof.
6.4
0
Existence results for large spatial domains
+
This section is devoted to the behaviour of the point spectrum of T K when the spatial domain R gets large. Actually we are concerned with
u(T
+ K ) n {A > - inf a ( . ) ) .
We assume that R = kR1 where R1 c Rn is convex and k E R is intended to go to infinity. We observe that the spectral problem
is equivalent to
H(X,k)$ = a$
, $ E L2(R1 x V )
161
Chapter 6. Form positive collision operators
where 'Ij;(x, v) = ', k) is positive compact and that
2 r r (O"(v) + >.) 1v'K~(w, v)1 (H(>., k)'Ij;, 'Ij;) = dw dJ-L(v) "(v;.W)2 + (O"(v) + >.)2 JRn
(6.19)
Jv
Inl
'Ij;(x, v)e-ix.wdx. Let P1(>', k) 2: P2(>', k)··· where ~(w, v) = (27r)-n/2 be the eigenvalues of H(>', k) . We begin with the basic result Lemma 6.3 (i) Pm(>', k) < liSA II for all mEN, k E R, >. > - inf 0"(.). (ii) For all "X > - inf 0"(.) and mEN, Pm(>', k) --+ IISAII uniformly in
>. E ["X, oo[ as k --+
00.
Proof (i) It follows, from (6.19) and Parseval's identity, that \ k)·I'1',.•'I'1.) L2(nlXV} (H( A,
=
J J J J
=
IIMAv'K~1I2 ~ IISAIIII'Ij;11 2.
<
Rn
Rn
dw
dw
V
dll.(v) 1v'K;;(w.v} r~ 12 yU(V}+A
v dJ-L(v)
IMAv'K~(W,v)12
This shows the first claim. To prove (ii), we choose arbitrarily
By the max-min principle,
Pm(>', k) 2:
min (H().., k)f, J) = (H(>., k)f(>., k), f().., k)) .11111=1
!EXm.
where
f(>.,k)(x,v) Hence
= 'Ij;(>.,k)(x)
'Ij;(>.,k)
E
Fm, 11'Ij;(>.,k)II£2(nd
= 1.
Topics in Neutron Transport Theory
162
Letting kj ~ 00, the sequence {-lP(>', kj);j E N} is relatively compact and we may assume that
'ljJ(>., kj ) ~ 'ljJ(>.) E Fm , Hence (i;(>., kJ· )
~
(i;(>.) in L2(Rn) and
11'ljJ(>.) II = 1.
II(i;(>.) I
1.
L2(Rn)
By Fatou's
lemma we deduce, from (6.20), lim inf Pm(>' , kj) 2: k;--+oo
2
ivrdJL(v) 1 J a(v )+ >. 1 JKcp(v)
Since cp is arbitrary,
limki~ooPm(>.,kj) 2: IIM"JKI1
2
=
J
IIS"II·
Hence, in view of (i),
and finally It is not difficult to see that the convergence is uniform in >. E [X, 00 [. <> We are in a position to prove the main result of this section. We define the size of 0 as the radius of the greatest ball included in O. We have
Theorem 6.8 a(T + K) n {>. > - inf a(.)} i:- 0 for 0 large enough if and only if a(B) n {A; >. > - inf a(.)} i:- 0. In such a case, the number of eigenvalues of T + K increases indefinitely with the size of 0 and all these eigenvalues converge to X the leading eigenvalue of B . Proof There is no loss of generality in assuming that 0 = kO l where 0 1 is fixed. According to the observations before Lemma 6.3, >. > - inf a(.) is an eigenvalue of T + K if and only if 1 is an eigenvalue of H(>', k) in L2(01 X V) . According to Lemma 6.1,
a(B) n {A; >.
> - inf a(.)}
=
0
if IIS"II < 1 for all >. > - inf a( .). In such a case, according to Lemma 6.3, IIH()., k)1I < 1 for all k E R and all >. > - inf a(.) so that a(T +
163
Chapter 6. Form positive collision operators
K) n {>. > - inf a(.)} is empty regardless of the size of n. Conversely, if a(B) n {>.; >. > - inf a(.)} =f. 0 then, according to Lemma 6.1, >. > -infa( .) -->
IIS,\II
is strictly decreasing
IISrII
and B has a leading eigenvalue 'X defined by = 1. Let /3 < 'X be arbitrary and let mEN be arbitrary also. Then, according to Lemma 6.3, Pm(>. , k) --> IIS,\II as k --> 00 uniformly in [/3,'X]. Therefore
PI (/3, k)
~
P2(/3, k)
~
...
~
Pm (/3, k) > 1 for k large enough
and finally there exist /31, /32, ... , /3m E ] /3, 'X [ such that Pi (/3i, k) = 1 , i = 1, 2, ... , m. This ends the proof. <> Remark 6.4 We already observed, in a more general setting, that the point spectrum of T + K is empty regardless of the size of n if the point spectrum of B is empty and that the converse result is not true (see Sections 5.5 and 5.6). Theorem 6.8 above shows that the converse is true for the model of this chapter. We note that the evenness assumptions (6.1) (6.2) exclude the counterexample pointed out in Remark 5.1.
6.5
The isotropic models
This section is devoted to additional spectral results pertaining to the assumption that the cross sections are isotropic, that is independent of the directions of velocities. To this end we assume that
dJ..L(v) := da(p)
@
dr(w) , P =
lvi,
v w= P
where da(p) is a Radon measure on [a , b] (0 ~ a < b ~ 00), and dr(w) is the surface Lebesgue measure on the unit sphere of Rn . Here,
Acp
8cp -a(p)cp(x,p,w) + = -PW'-8 x
lb
I
I
k(p,p )da(p)
a
Is
and
is self-adjoint compact. We observe that the spectral problem
Acp = >.cp; Re>. > s(T)
I
I
I
cp(x,p ,w )dr(w)
Sn-l
Topics in Neutron Transport Theory
164 amounts to 1
cp(x, p, w} = -
p
l
S(x,W)
e
('>'+u(p»' P
.
K1/J(x - sw, p}ds
(6.21)
0
where
hn-,
1/J(x,p} =
cp(x, p, W'}dT(W').
Further, for convex 0, (6.21) is equivalent to _ (.>.+,,(p»
1/J(x,p} =
1 n
dx
,e
1",-/1
p
pix - x
,
2
,
n-1
I
K1/J(x ,p}; 1/J E L (0 x [a,b]) .
(6.22)
The main feature of isotropic models is the following Theorem 6.9 Let the collision operator be self-adjoint and isotropic. Then the point spectrum of A in the half plane {ReA> - inf a( .}} is real. Proof. Suppose there exists a nonreal eigenvalue A = A1
+ iA2
(A2
¥
O). We define
E A : cp E L2 (0 x [a,b])
-t
1
e
1 1
(.>.+u(p»
x-x'
p
n pix - x
, n-1
I
cp(x " ,p}dx .
The spectral problem (6.22), i.e.
implies, in view of the self-adjointness of K, that
(E AK1/J,K1/J) is real (K1/J
¥ O) .
On the other hand, using Fourier transform with respect to the space variable and Parseval's identity,
where
165
Chapter 6. Form positive collision operators and
K;fi(w,p) =
(27r~n/2l K7/J(x,p)e- iw .x dx .
Hence
and therefore
because
Im(~(p, w)) =0
(6.23)
IK7/J12 (w , p) > O. On the other hand, 00
~(p, w) = ~ / e_ c>.+upCp»r dr / o
1/
Sn-l
e- ~ p dr
o
Let us set
p(s) =
/ 1
00
p
e-irw'. wdr(w' )
e-,·rw8 11 ds
-1
L,.f,j;r
=8
/ WI .
dr(w'); s
E
1:1=5
[-1 , 1]
{w'
the (n - 2)-dimensional measure of the set E sn-l; note that p(±I) = 0 and that p(.) is even. Then
w' . 1:1 = s} . We
~(p,w) 1
p(s) ds / A+O"(p)+iplwls
-1 1
/ [A + O"(p/+ ip IWI s o 1
2/3J o
+ >. + O"(p/- ip Iwl s Jp(s)ds 00
p(S) ds=_2_J p(I/t) /3 dt /32 + p21wl 2 S2 p IWI (-.1!1..)2 + 1 p Iwl (6.24) 1
plwl
where /3 = A + O"(p) . Let /3 = /31 + ifh. Using the fact that /31 > 0 and integrating by parts, one sees that the imaginary part of ~(p, w) is equal
Topics in Neutron Transport Theory
166
11
to
[(1- ~)2 + (~)2l
00
-2plwll
I
p(l/t)Log (1+~)2+(~)2
dt t2 '
Finally, since pi (lit) < 0,
~ ) { < 0 for Im'\ > 0 1m E>.(p, w) > 0 for Im.\ < 0
(
and this contradicts (6.23). <> For the rest of this section, we assume that K is positive in the scalar product sense. Then.\ is an eigenvalue of A if and only if 1 is an eigenvalue of the self-adjoint compact operator in L2(n x [a, b])
H>.
:=
.JKE>..JK.
On the other hand,
(H>.1f;,1f;) =
lb kn da(p)
E;,(p,w)
1.JK1f;1
2
(w,p)dw
and, by (6.24), for real.\
E;,(p, w) = 2(.\ + O'(p))
r p(s) Jo (.\ + a(p))2 + p21wl 1
2
s2
ds > O.
(6.25)
Thus H>. is positive. We define the operator
- lb K :
a
k(p,p') r7
VPVP
I
r.
As in the previous sections, K = [,}..JK] [,}..JK
We observe that K
is bounded if and only if ,}..JK is. We assume, for the sequel, that 0 E V
(a = 0) and that 0'(0) = inf 0'(.). If K is bounded, then H>. has a limit in the operator norm as .\ ~ - 0'(0)
where
0-<7(0) :
x [a, b]) ~
In
(,,(p)-,,(O»lx-X/I
e
Ix _ x:l n - 1
167
Chapter 6. Form positive collision operators
Theorem 6.10 Let K be bounded in L 2 ([a, b]) . Then O'(A)n{Re>. > -O'(O)} consists of m real eigenvalues where m is the number of eigenvalues of H -<1(0) exceeding one. Proof Let 1I1(>') ;::: 1I2(>')'" be the eigenvalues of H>.. Arguing as in Remark 6.3, it suffices to prove that the curves
>. - t lIi(>')
i = 1,2, ...
are strictly decreasing. To this end it suffices that H>. - H{3 (>. < (3) be a positive operator in the scalar product sense. This is true if
>. - t E>.(p,w) is stricly decreasing. By integrating by parts the last integral in (6.24), E>.(p,w)
2
= plwl
[{OO I
p (l/t)tan-
il
(>.+O'(p))t
1
plwl
(
)
dt t2
7r
+ '2 P(O)
1
which ends the proof. 0 We complement Theorem 6.10 with a converse result. Theorem 6.11 We assume that K is not bounded in L2([0, b]). O'(A) n {Re>. > -O'(O)} consists of infinitely many real eigenvalues.
Then
Proof It suffices to prove that
lim
>. ..... -<1(0)
Let
= 1)
lIi(>')
=
+00 (i
= 1,2, ... ).
be such that
J:.....jK
v:
(6.26)
We choose an arbitrary integer m. Let Fm C L2([2) be an m-dimensional subspace and define
Xm
:=
According to the max-min principle,
where
Fm I8i <po
168
Topics in Neutron Transport Theory
Hence the lower estimate of lIm(A)
b
J
da(p)
o
~ 12/ luA(w)1 - 2 dw p Iwl 2 v Kcp(p)
1
J00
.
~
P(rplwl) r2 + 1 dr
(6.27)
~
where (3 = A + a(p). Let Aj -. -a(O) be a decreasing sequence. By the compactness of the unit sphere of Fm , we may assume that UAj - . U
in Fm ,
lIuli = 1.
By passing to the limit in (6.27), Fatou's lemma yields
= 2.[ 0b da(p) 1Jp Vi< cp(p) 12 /
Rn
lu( w) 12 dw I!I /00 Jr
p
(L) 7Ji!;1 dr
where if = u(p)-u(O) . By using the assumption sup u(p)-u(O) = p p the fact that p( .) is decreasing on [0,1] we obtain
C
< 00 and
limAj--+-U(O) lim (Aj)
in view of (6.26) . <>
6.6
Comments
The theory above unifies and extends previous works by the author [9] [11] devoted, respectively, to the continuous and multigroup models. We restricted the analysis to homogeneous cross-sections although many results
169
Cbapter 6. Form positive collision operators
still hold for nonhomogeneous scattering kernels since H>. remains selfadjoint. However, the evenness assmnption (6.2) with respect to each variable is crucial. It would be very useful, but not so easy, to extend the theory above by replacing the strong evenness assmnption (6.2) by the more physicalone k(-v,-v') = k(v,v'). We point out that Theorem 6.4 admits a straightforward version for nonconstant collision frequencies. For isotropic models (see Theorem 6.10), the nmnber of eigenvalues of the transport operator is equal to the nmnber of eigenvalues exceeding one of the compact self-adjoint operator H -<7(0)' If the latter is Hilbert-Schmidt or has, at least, some Hilbert-Schmidt power, then one can derive a very explicit upper bound of the nmnber of eigenvalues of the transport operator (see [9]). We point out that it is possible to estimate the distances between two successive eigenvalues of the transport operator if the curves Pm(.) are decreasing, in particular for the isotropic models (see [9]). ill Chapter 5, several results on the existence of eigenvalue are given. The first ones, when the velocity space is bounded away from zero, are tied to irreducibility argmnents, while those of the present chapter (Theorem 6.6) rely on the assmnption that the collision operator is positive for the inner product. The respective assmnptions are of very different nature and it seems mathematically interesting to analyze further their possible connections. We point out, in passing, that it is possible to use the precise existence results of this chapter, to prove existence results for more general scattering kernels by comparison argmnents as in [8] . According to Theorem 6.6, the transport operator has infinitely many real eigenvalues accmnulating at -00 if the collision operator is not degenerate. Actually this result is not optimal since, for monoenergetic models with one rank collision operators, the transport operator still has infinitely many real eigenvalues (see R.V. Norton [14] and S. Ukal [17]). The reality result (Theorem 6.9) was first proved for a simple monoenergetic model by J. Lehner and M. Wing [5] and extended to the continuous models by S. Albertoni and B. Montagnini [1] and J. Mika [6]. It is very strange that, untill now, the only tool for proving the reality of the point spectrum, in the half-space {A; ReA > - inf a(.)}, is that introduced by J. Lehner and M. Wing [5]. This explains why the problem of the reality of the point spectrmn for nonisotropic models is completely open. We point out, however, that this technique extends to very special separable nonisotropic models' for instance in slab geometry, if k(p., p. ) = f(p.)g(p. ) and if h(p.) = f(p.)g(p.) is even in [-1,1] and is decreasing on [0,1] then the point spectrmn is real (see M. Mokhtar-Kharroubi [10]). It is possible, however, in quite general settings, to find explicit curves in the complex plane, delimiting nontrivial regions I
I
"
170
Topics in Neutron llansport Theory
where no complex eigenvalue may exist (see [lo]). The reality of the point spectrum is, probably, one of the most challenging mathematical problems in neutron transport theory. We point out that the existence of nonml eigenvalues in the opposite half-space {A; ReX < - inf a ( . ) )was proved by B. Montagnini and V. Pierpaoli [12], B. Montagnini [13], Xu Bang Qing, Xao Chang Gui and Wang Yong Jiu [19] and M.Yang, Z. Kuang and X. Wang [21]. The fine description of the spectrum of transport operators, beyond direct consequences of compactness results, started with the classical (and very nice) paper by J. Lehner and M. Wing [5] and was continued, in the framework of isotropic models, in particular by G.H. Pimbley [15], R.V. Norton [14], S. Albertoni and B. Montagnini [I],J. Mika [6][7], H.G. Kaper [4] and S. Ukai [17][la]. The use of the square root of collision operators, in the framework of isotropic models, goes back to S. U k i [la]. The extension to continuous or multigroup nonisotropic models, under suitable evenness assumptions, is due to M. Mokhtar-Kharroubi [9][ll]. Fine spectral results for model transport operators in reflected spheres are given by D.C. Sahni and N.G. Sjostrand [16]. Finally, we mention a paper by Xu Genqi, Yang Mingzhu and Wang Shenhua [20],on transport operators with velocities bounded away from zero, where a summability of the eigenvalues of transport semigroups (for large times) is given. This paper gives a nice contribution to a classical problem on the eigenfunction expansion of transport sernigroups, in the spirit of B. Montagnini and V. Pierpaoli's paper [12] on the rod model.
References [I] S. Albertoni and B. Montagnini. On the spectrum of neutron transport equation in finite bodies. J. Math. Anal. Appl. 13 (1966) 19-48.
[2] M.S. Birman and M.Z. Solomjak. Spectral Theory of Self-adjoint Operators in Hilbert Space. D.Reide1 Publishing Company, 1987. [3] H. Brezis. Analyse Fonctionnelle: Theorie et Applications. Masson, Paris, 1983. [4] H.G. Kaper. The initial-value transport problem for monoenergetic neutrons in an infinite slab with delayed neutron production. J. Math. Anal. Appl. 19 (1967) 207-230. [5] J. Lehner and M. Wing. On the spectrum of an unsyrnrnetric operator arising in the transport theory of neutrons. Comm. Pure Appl. Math. 8 (1955) 217-234.
Chapter 6. Form positive collision operators
171
[6] J. Mika. Time dependent neutron transport in plane geometry. Nucleonik. 9 Bd, Helft4 (1967) 200-205. [7] J. Mika. The effects of delayed neutrons on the spectrum of the transport operator. Nucleonik. 9 Bd, Helftl (1967) 46-51. [8] M. Mokhtar-Kharroubi. Some spectral properties of the neutron transport operator in bounded geometries. Transp. Theory Stat. Phys. 16(7) (1987) 935-958. [9] M. Mokhtar-Kharroubi. Spectral theory of the neutron transport operator in bounded geometries. Transp. Theory Stat. Phys. 16(46) (1987) 467-502. [10] M. Mokhtar-Kharroubi. Quelques remarques sur les valeurs propres complexes de l'operateur de transport des neutrons. Publications mathematiques de Besan{:on, 1988-1989. [ll] M. Mokhtar-Kharroubi. Spectral theory of the multigroup neutron transport operator. Eur. J. Mech. B/ Fluids. 9(2) (1990) 197-222. [12] B. Montagnini and V. Pierpaoli. The time-dependent rectilinear transport equation. Transp. Theory Stat. Phys. 1(1) (1971) 59-79. [13] B. Montagnini. Existence of complex eigenvalues for the mono-energetic neutron transport operator. Transp. Theory Stat. Phys. 5(2-3) (1976) 127-167. [14] R.V. Norton. On the real spectrum of a monoenergetic neutron transport operator. Comm. Pure Appl. Math. 15 (1962) 149-158. [15] G.H. Pimbley. Solution of an initial value problem for the multivelocity neutron transport equation with a slab geometry. J. Math. Mech. 8 (1958). [16] D.C. Sahni and N.G. Sj6strand. Time eigenvalues for one-speed neutrons in reflected spheres. Talk given in 15th International Conference on Transport Theory, 1-7 June 1997, G6teborg. [17] S. Uka.L Real eigenvalues of the monoenergetic transport operator for a homogeneous medium. J. Nucl. Sci. Technol. 3 (1966) 263-266. [18] S. ukai. Eigenvalues of the neutron transport operator for a homogeneous finite moderator. J. Math. Anal. Appl. 30 (1967) 297-314.
172
Topics in Neutron Transport Theory
[19] Xu Bang Qing, Xao Chang Gui and Wang Yong Jiu. Existence of infinitely many complex eigenvalues for the monoenergetic transport operator. Lett. Math. Phys. 11 (1986) 95-105. [20] Xu Genqi, Yang Mingzhu and Wang Shenhua. On the eigenfunction expansion of the transport semigroup for a bounded convex body. Transp . Theory Stat. Phys. 26{1 & 2) (1997) 271-278. [21] M. Yang, Z. Kuang and X. Wang. Complex eigenvalues of a monoenergetic neutron transport operator. Transp. Theory Stat. Phys. 26{1 & 2) (1997) 253-261.
Chapter 7
On Miyadera perturbations of • Co-semlgroups 7.1
Introduction
This chapter is devoted to a perturbation theory of co-semigroups for a class of unbounded perturbations referred to as Miyadera perturbations. This class was introduced by I. Miyadera [2] in the sixties and investigated later by J . Voigt [5] [6] [7], A. Rhandi [4] , W. Arendt and A. Rhandi [1] and M. Mokhtar-Kharroubi [3] . It is a very natural (but not trivial) extension of the standard bounded perturbations. Miyadera perturbations have enjoyed a renewal of interest in the context of positive semigroups on L 1 (,.,,) spaces as will be shown in Chapter 8. Moreover, they provide us with a useful framework for singular neutron transport equations involving unbounded collision frequencies and unbounded collision operators in L 1 (,.,,) spaces as will be shown in Chapter 9. Let us explain the contents of this chapter. Let X be a complex Banach space and let {U(t) jt ~ O} be a eo-semigroup with generator T. We denote by D(T) the domain of T . We consider a T-bounded linear operator B, i.e.
B E L(D(T)j X) 173
(7.1)
174
Topics in Neutron n i t ~ s p o r Theory t
where D(T) is equipped with the graph norm. We say that B is a Miyadera perturbation of T if there exist 0 < a < oo and 0 5 y < 1 such that
We are interested here in two problems: (i) Does T
+ B generate a Q-semigroup in X
(ii) Compare the essential types of {et(T+B);t
> 0)
and {etT; t
> 0) .
The two questions are answered by J. Voigt [5] [7] and A. Rhandi [4]. The purpose of this chapter is to present some of their results.
7.2
A perturbation theorem
We define inductively (and formally for the time being) the operators Uo(t) = U (t) and
The main result of this section is the following
Theorem 7.1 (J. Voigt [5] [7] , A. Rhandi [4]). The operators Uj (t) ( j 2 1) are bounded on X. The operator A = T B with domain D(A) = D(T) is a generator of a Q-semigroup {V(t); t 0)) given by a Dyson-Phillips Uj (t) where the series converges in L ( X ) uniformly expansion V(t) = in bounded times.
xco
+ >
Proof: Our presentation is slightly different from the original ones. Let x E D(T). Then
175
Chapter 7. On Miyadera perturbations of co-semigroups and
IIU1 (t)xll :::; c(t)
J:
IIBU(s)xll ds (c(t) = sUPrE[O,tjIlU(T)ID
[iJ
J(Hl)a
:::; c(t) L j : o
.
IIBU(s)xll ds ([~] is the integer part of~)
Ja
Ja° IIBU(r + ja)xll dr = c(t) L j[iJ: o Ja° IIBU(r)U(ja)xll dr
[iJ
= c(t) L j : o
:::; I'c(t) L;:~ IIU(ja)xll
:::; c(t) Ilxll
where c(t) = I'c(t) L:~~b IIU(ja) II· Hence U1(t) extends in a unique way to the whole space X as a bounded operator and
t
E [0, oo[ -+
U1(t)
E L(X) is locally bounded.
It follows, by induction, that
and that
t
E [0, oo[ -+
Uj(t)
E L(X) are locally bounded.
(7.4)
On the other hand, one shows, by induction, that
t
E [0, oo[ -+
Uj(t)x
E
X are continuous when x
E
D(T) (j
~
1).
Hence, in view of (7.4), it follows, by density, that
t
E [0, oo[ -+
Uj(t)
E L(X) are strongly continuous
(j ~ 1) .
(7.5)
Let us show the following identities n
LUj(t)Un-j(S)=Un(t+s) nEN t,s~O. j=O
(7.6)
Topics in Neutron Transport Theory
176
o.
Clearly (7.6) is true for n = for x E D(T),
=
Assume it is true for some integer n. Then,
sUn(t + s - T)BU(T)xdT + Jt+s Un(t + s - q)BU(q)xdq J s
0
Hence n+l
L U (t)Un+l- (s)x = Un+l(t + s)x j
j
x E D(T)
j=O
and, by density, (7.6) is true for n + l.We deal now with the convergence of the series Ei=o Uj(t) . We assume that
IIU(t)11 ::; Me wt
; M
2: 1, w 2: 0
and note that if t E [0, a] and x E D(T) then, in view of (7.2),
IIU1(t)xll ::;
J:
: ; J:
Mew(t-s)
IIBU(s)xllds
Mew(t-s)
IIBU(s)xll ds ::; "(Mewt
Ilxll·
One sees, by induction, that (7.7) so that 00
L IIUj(t)11 converges uniformly in t E [0, a] j=O
(7.8)
Cbapter 7. On Miyadera perturbations of eo-semigroups and
177
00
L Uj(t) converges in L(X) uniformly in t E [0,0'] . j=O Ai; a consequence of (7.8), the product series 00
L Wn(t) converges in L(X) uniformly in t E [0,0'] n=O where
n
Wn(t) = L Un-j (t)Uj (t) . j=O In view of (7.6), this amounts to 00
L U (2t) converges in L(X) uniformly in t E [0,0'] , n
n=O i.e.
00
L Un(t) converges in L(X) uniformly in t E [0,20'] . n=O By iterating the process one sees that the series converges for all t uniformly in bounded times. Let 00
Vet) = L Un(t) . n=O The identities (7.6) show that
L~=o Un(t + s)
=
L~=o L;=o Uj(t)Un_j(S)
=,,~ Uj(t) , , 0 0 . Un-j(s) L..-J=o L..-n=J
i.e.
Vet
+ s) = V(t)V(s) .
Moreover, the convergence being locally uniform in time, t E [0, oo[ --+
V(t) E L(X) is strongly continuous.
~
a
Topics in Neutron am port Theory
178
Thus, {V(t);t 2 0) is a co-semigroup satisfying the Duhamel equation
It remains to identify its generator A. To this end we need the following
We proceed as follows
This shows that (7.10) is true and, since y < 1,
11B(X- T)-'11 < 1 for X large enough.
(7.11)
By taking the Laplace Transform of the Duhamel equation (7.9) and using fibini's Theorem, thanks to (7.10), we obtain
and therefore
In view of (7.11) , [I- B(X - T)-'1 is an isomorphism whence (A - A)-' and (A - T)-' have the same range, i.e. A and T have the same domain. F'inally, coming back to the Duhamel equation (7.9) and noting that
179
Chapter 7. On Miyadera perturbations of eo-semigroups
because U(s)x
-+
x in D(T), one sees that
c 1 lot V(t -
s)BU(s)xds
-+
Bx as t
-+
0,
i.e. Ax = Tx + Bx. 0 We close this section with some estimates we shall need later
Lemma 7.1 Let w = w(U) be the type of {U(t); t 2: O}. Then, for each integer j 2: 0 and w' > w(U) , there exist Cj(w') such that IIUj(t)11 :=:; cj(w')ew't. Proof: The estimate is clear for j = O. Assume it is true for j = n. Starting with
Un+!(t)x = lot Un(t - s)BU(s)xds ; x E D(T) and taking w(U) < w" < w' , we obtain
IlUn+!(t)xll :=:; J: cn(w")e(t-s)w" IIBU(s)xll ds :=:; cn(w")e tw
"
[.!oj J(k+l)<> sw" L"k=O eIIBU(s)xll ds k<>
I
tw
"
II
:=:; cn(w ,," )e sUPTE[O,<>J e- w"T ,,",[*J .LJk=O J<>0 BU(T)e- W k<>U(kO'.)x dT
where " h sup e- w" [ [ -t ] cn+! (w , ) = en(w" )eo(w T sup TE [O ,<>J
This ends the proof.
0
t~O
0'.
, + 1] et(w"-w).
180
7.3
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The essential type of Miyadera perturbations
The main result in this section is the following Theorem 7.2 Let {U(t)j t :2: O} be a eo-semigroup with generator T and type (resp . essential type) w(U) (resp . we(U)) . Let B be a Miyadera perturbation ofT and {V(t)j t :2: O} be the eo-semigroup with generator A = T+B. Let there exist mEN and a sequence {tk} C [O,oo[ such that tk --+ 00 and Rm(tk) = V(tk) - 2:;',:01 Uj(tk) is compact (or even strictly singular if X is isomorphic to some V(IL) space) . Then re(V(t)) ::; re(U(t)), i.e. we(V) ::; we(U).
The proof is based on two technical results Lemma 7.2 Let K E L(X) be compact and let B
= B+K .
Let {Uj(t)jj :2:
be defined inductively by (7.3) where B is replaced by B . Then, for all to > 0 and j :2: 0, {Uj(t) - Uj(t)j 0::; t ::; to} is collectively compact. Proof: We recall that M C L(X) is said to be collectively compact if there exists a relatively compact subset of X containing A(Bx) for all A E M where B x is the unit ball of X . We prove the lemma by induction on j :2: 0 assuming it is true for j - 1. We note that, for x E D(T), Uj(t)x - Uj(t)x
=
=
J: It
.
+
[Uj-l(t - s)B - Uj_l(t - s)B] U(s)xds
Uj_l(t - s)(B - B)U(s)xds
0
J:
[Ui-l(t - s) - Uj-l(t - s)] BU(s)xds.
Let c = sup {I/U(s)11 j 0::; s ::; to}. Then (B - B)U(s)(Bx)
c
K(cBx)
j
0::; s::; to.
The strong continuity of Ui-l(.) implies C = {Ui-l(S)K(cBx)jO::; s::; to} is relatively compact
(7.13)
O}
181
Chapter 7. On Miyadera perturbations of co-semigroups
and
lt
Uj-1(t - s)(.8 - B)U(s)ds(Bx) C toconv(C) ; t::; to.
(7.14)
On the other hand, by the Miyadera condition, there exists "(' 2: 0 such that
l
ta
IIBU(s)xll ds ::; "('
Ilxll ;
x E D(T).
Let
d = {(Uj_1(s) -
Uj_1(s)) (Bx);O::; s::; to}
which is relatively compact by the induction hypothesis. Let x E D(T), Ilxll ::; 1 and 0 ::; t ::; to. If x' E X' is such that
:~g, 1( x' , y) 1 ::; 1 (i.e. x' E (C')O the polar set of C' ) then
1( x' ,.f: [Uj-l(t -
: ; J: I(
s) - Uj-l(t - s)] BU(s)xds) 1
x' , [Uj_1(t - s) - Uj_l(t - s)] BU(s)x)1 ds
::; J>BU(S)X II ds ::; "(' . According to the bipolar theorem ([8] Theorem 2, p. 137), we deduce
lt
[Uj_1(t - s) - Uj_1(t - s)] BU(s)xds
E "(' conv(C ' ).
(7.15)
Finally, by combining (7 .14) and (7.15), we obtain
(Uj(t) - Uj(t)) (Bx) C toconv(C) and then {(Uj(t) - Uj(t)) (Bx);O::;
+ "(' conv(C' ) ,0::; t ::; to
t::; to}
is relatively compact.
<>
Lemma 7.3 Let w' > we(U) and n E N. Then there exists a constant en(w ' ) such that n
re(LUj(t)) ::; cn(w')etw' j=O
(t 2: 0) .
Topics in Neutron '1Tansport Theory
182
Proof Let PI be the projection corresponding to O"(T) n {A; ReA :2: w' } . We note that PI is a finite rank projection commuting with {U(t); t :2: O} and the type of {U(t)IXe; t :2: O} is < w' where Xe = Pe(X) and Pe = I - PI . In particular, there exists M:2: 1 such that
The subspaces XI = PI(X) ; Xe = Pe(X) are invariant under {U(t); t :2: O} and XI C D(T) (dimXI < 00). It follows that
Pe E L(D(T); D(T)) . Let
-
-
13 = PeBPe. Then 13 E L(D(T);X)
.I:
IIBU(t)xll dt
J: :.:; J:
=
and, for x E D(T),
IIPeBPeU(t)xll dt
=
J:
IIPeBU(t)Pexll dt
IIBU(t)Pexll dt :.::; 'Y IIPex11
:.: ; 'Y Ilxll·
Hence 13 is a Miyadera perturbation of T . We denote by Uj(t) (j:2: 0) the corresponding iterations. It follows from
that Uj(t) leaves Xe invariant and vanishes on XI' Moreover, the parts of
Uj(t) on Xe are nothing but the iterations corresponding to U(')IXe and the Miyadera perturbation PeBIXenD(T)' According to Lemma 7.1, there exists a constant en (w') such that
Since (Xe , XI) is a pair of reducing subspaces for 00, it follows that
r.
[t,
U;
2:7=0 Uj(t) and dim XI
(t)]-; c,,(w' )e"'" .
<
(7.16)
Chapter 7. On Miyadera perturbations of co-semigroups
183
To prove the lemma, it suffices to show that
We prove this in two steps. First let B = BPe; it is a Miyadera perturbation of T . Let Uj(t) (j :2: 0) be the corresponding iterations. Since B - B = -BPj is compact, it follows from Lemma_7.2 that Uj(t) - Uj(t) is compact for t :2: 0 and j :2: O. On the other hand,
B = PeB so that
~ne shows, by induction, that Uj(t) commutes with Pe. It follows that
Uj(t) = PeUj(t) whence
-
Finally Uj(t) - Uj(t)
-
= (Uj(tt- Uj(t)) + (Uj(t) - Uj(t)) is also compact,
so that ,£j=o Uj(t) and ,£j=o UAt) have the same essential spectral radius which ends the proof in view of (7.16) . 0 Proof of Theorem 7.2 : Let w' > we(U). According to Lemma 7.3, there exists a constant Cn-l(W ' ) such that n-l
re(~=Uj(tk)):S Cn_l(w')ew'tk
V k.
j=O
Moreover, the assumptions of the theorem ensure that n-l
re(V(tk)) = re(L Uj(tk)) j=O
whence
i.e.
e(we(V)-w')t k :S Cn-l(W ' ) V k which shows that we(V) :S w' for all w' > we(U) . 0
184
7.4
Topics in Neutron Transport Theory
Comments
The generation theorem for Miyadera perturbations was given first (of course!) by 1. Miyadera [2] and is stated there in a different form. The statement we give is taken from J. Voigt [5] where it is shown, in particular, that the perturbed semigroup is given by a Dyson-Phillips expansion (as for bounded perturbations) for small times. The convergence of the Dyson-Phillips expansion for all times as well as the estimates of the terms ofthe series (Lemma 7.1) are due to A. Rhandi [4] who also proved that the essential type of the perturbed semigroup is less than or equal to the type of the unperturbed semigroup. The inequality between the essential types (Theorem 7.2) is due to J . Voigt [7] and actually holds under more general assumptions. We point out that the stability of the essential type for (unbounded) Miyadera perturbations is an open problem; the difficulty being that B (or-B) is not (apparently) a Miyadera perturbation of A = T + B. Finally, a useful open problem is to give sufficient conditions (in terms of the unperturbed semigroup and the perturbation B) implying the assumptions of Theorem 7.2.
References [1] W . Arendt and A. Rhandi. Perturbations of positive semigroups. Arch. Math. 56 (1991) 107-119. [2] 1. Miyadera. On perturbation theory for semigroups of operators. Tohoku. Math. J. 18 (1966) 299-310. [3] M. Mokhtar-Kharroubi. Characterisation BV des perturbations de Miyadera et applications. Workshop "Evolution equations, Control theory, Biomathematics". Luminy, March 8-12, 1993. [4] A. Rhandi. Perturbations positives des equations d'evolution et applications. These, Universite de F'ranche-Comte Besanc:;on, (1990). [5] J. Voigt. On the perturbation theory for strongly continuous semigroups. Math. Ann. 229 (1977) 163-17l. [6] J. Voigt. On substochastic CO-semigroups and their generators. Semes terbericht Funktionalanalysis, Thbingen, Wintersemester, (1984/85). [7] J. Voigt. Stability of the essential type of strongly continuous semigroups. Trans. Steklov. Math Inst. 203 (1994) 469-477. [8] K. Yosida. Functional Analysis. Springer Verlag, 1978.
Chapter 8
On resolvent positive operators and positive co-semigroups in Ll (J-L) spaces 8.1
Introduction
The aim of this chapter is to present some mathematical results about Desch's perturbation theorem [5] . This theorem emphasizes the particular role of positivity in perturbation theory in L1 spaces. Besides its mathematical interest, this perturbation theorem proves useful for the treatment of transport equations with singular cross-sections (i.e. unbounded collision frequencies and unbounded collision operators) which are considered in Chapter 9. This remarkable role of positivity, tied to the additivity of the L1 norm on the positive cone, appears also in the context of scattering theory in Chapter 12 and in the mathematical analysis of singular transport equations in Chapter 9. Let us state Desch's result [5] . TheoreIn 8.1 Let {U(t)j t :?: O} be a positive co-semigroups in L1(p,) with generator T and let B E L(D(T) j £1 (p,)) be a positive operator. Let there exist>. > s(T) (the spectral bound ofT) such that (>.-T-B)-l exists and is positive. Then T + B generates a (positive) co-semigroup.
We point out that this result is not true in V(p,) spaces (1 < p < 00) [2]. We do not present Desch's proof but rather different proofs which are 185
186
Topics in Neutron Transport Theory
somewhat simpler than the original one and provide useful informations on the perturbed semigroup in preparation for the spectral analysis of singular transport equations we deal with in Chapter 9. The first proof we give relies on repeated applications of the Miyadera perturbation theorem described in Chapter 7. The second one, based directly on Miyadera perturbation theorem, relies on renorming arguments. Finally, we provide a third alternative proof independent of the Miyadera perturbation theorem.
8.2
A preliminary result
Let X be a Banach lattice and T be an unbounded linear operator acting on X with domain D(T). We recall that the spectral bound of T is defined as s(T) = sup {ReA; A E a(T)} . We say that T is resolvent positive if s(T) < for A > s(T). We start with
00
and (A - T)-l is positive
Lemma 8.1 Let X be a Banach lattice, T be a resolvent positive operator and A > s(T) . Let B E L(D(T); L1(J.L)) be a positive operator. Then the following conditions are equivalent. (i) r 17 (B(A - T)-l) < 1. (ii) A E p(T + B) and (A - T - B)-l is positive.
Proof: Let A > s(T) be such that r 17 (B(A - T)-l) problem AX - T x - Bx = y ; X E D(T).
<
1. Consider the
It is clearly equivalent to X -
(A - T)-l Bx = (A - T)-ly
X
E D(T) .
Letting Bx = z, (8.1) amounts to z - B(A - T)-lz
= B(A -
T)-ly z E X
and therefore 00
z = (I - B(A - T)-l)-l B(A - T)-ly = ~)B(A _ T)-l)n+l y . n=O
Finally
(8.1)
Chapter 8. On positive co-semigroups in L 1 (J.L) spaces
187
Hence 00
(>. - T - B)-l = (A - T)-l ~)B(A - T)-l)n ~ (A - T)-l ~ o. n=O Conversely, let A > s(T) be such that (A-T-B)-l ~ O. Then the identity n
(A - T - B}2:(A - T)-l(B(A - T)-l)j = I - (B(A - T)-l)n+l j=O
implies
n+1 2:(B(A-T)-l)j = B(A-T-B)-l(I _(B(A_T)-1)n+1) ::; B(A-T-B)-l j=l whence rq(B(A - T)-l) ::; 1 and, for any J.L > 1, [J.L - B(A - T)-l] -1
=
f
(B(A ;~)-1 )j ::; I
+ B(A -
T _ B)-l
j=O
so that
On the other hand (see, for instance, [4] Proposition 2.19, p. 96),
because the spectral radius of a positive operator belongs to its spectrum (Theorem 5.1) and therefore (8.2) shows that rq(B(A - T)-l) < 1. 0
8.3
Miyadera perturbations in L 1 (J.L) spaces
We start with a weaker version of Desch's result. Lemma 8.2 Let T be the generator of a positive co-semigroup in L 1(J.L) and let BE L(D(T); L 1(J.L)) be a positive operator. We assume there exists A > s(T) such that IIB(A - T)-lll < 1. Then T + B generates a (positive) co-semigroup.
Topics in Neutron Transport Theory
188
Proof: Let x E D(T), x ~ O. Then, because of the additivity of the U norm on the positive cone,
J~ IIBe-AtU(t)x ll dt
= liB
J~ e-AtU(t)xdt ll
= IIB(A -
T)-lxll
::; IIB(A - T) -lllllxli whence, for all a:
> 0,
lD.IIBe-AtU(t)XII dt::; 'Y Ilxll
; x E D(T)
, x
~0
where 1
1
Let x E D(T) and x~ = n Jon U(t)x+dt , x:: = n Jon U(t)x- dt where x+ (resp. x-) is the positive part (resp. the negative part) of x. It is easy to see that
x~
-
x~
-->
x in D(T) as n
--> 00.
(8 .3)
From
we pass to the limit , thanks to (8 .3) , and get
i.e. B is a Miyadera perturbation of the generator T - AI. The generation result follows from the results of Chapter 7. <> We are now ready to prove
Theorem 8.2 Let T be the generator of a positive eo-semigroup in L 1 (J1.) and let BE L(D(T);U(J1.)) be a positive operator. We assume there exists A > s(T) such that (A - T - B)-1 is positive. Then T + B generates a (positive) eo-semigroup.
Chapter 8. On positive co-semigroups in Ll (J.L) spaces
189
Proof: According to Lemma 8.1, ru(B().. - T)-l) < 1. By replacing B by sB, s E [0,1]' Lemma 8.1 asserts that
Let n E N be such that
Then
In particular,
Thus, (8.4) allows us to apply Lemma 8.2 repeatedly for j = 0, 1, ... , n-l with the perturbation n- l B and this ends the proof. <> Because of the repeated use of the Miyadera perturbation theorem it is not clear, a priori, that the perturbed semigroup is given by a DysonPhillips expansion. To prove it is so we need the following result. Lemma 8.3 Let C E L(Ll(J.L)) be a positive operator such that ru(C) < 1. We denote by II the usual Ll norm. Then there exists an equivalent norm IlIon Ll(J.L) which is additive on the positive cone and such that the corresponding operator norm of C is less than one.
Proof: Let c and c' be two constants such that no E N be such that
°< c < c' < 1 and let
We set
(8.5) It is easy to see that 1.11 is a norm in Ll(J.L) and is additive on the positive cone. Moreover, it is clear that
190
Topics in Neutron Transport Theory
and
::::;
o ,,00 (~C)OIl)k "n ~p=O (c'Y ~k=O c' no Ixl
::::;
o .lJQ!]. ,,00 ( no ) "n ~p=O (c'Y ~k=O "Pro
-1.lJQ!].
-1
k
Ixl
which shows the equivalence of the two norms. Finally
whence
IIGll 1 ::::; c' < 1
where 11111 is the operator norm associated with 111'
0
Theorem 8.3 Under the assumtions of Theorem 8.2 the semigroup generated by T + B is given by the Dyson-Phillips expansion. Proof: We denote by II the usual P norm. According to Lemma 8.1, there exists). > s(T) such that r 17 (B(). - T)-l) < 1. In view of Lemma 8.3, there exists a norm 111 equivalent to II, additive on the positive cone and such that IIB(). - T)-1111 < 1. By using the new norm, the proof of Lemma 8.2 shows that B is a Miyadera perturbation of T - ),1 and, according to the general theory of Chapter 7, e-Atet(T+B) =
00 L: Uj(t)
(8.6)
j=O
where Uj+l(t) = lot Uj(t - s)Be-ASU(s)ds , Uo(t) = e-AtU(t) . The series (8.6) converges in the operator norm 11111 (and therefore in the usual operator norm) uniformly in bounded times. On the other hand, the convergence of the series (8.6) is obviously equivalent to
00
et(T+B) = L:Uj(t) j=l
Chapter 8. On positive co-semigroups in L 1 (JL) spaces
191
where
Uj+1(t) = lot Uj(t - s)BU(s)ds
j
Uo(t) = U(t)
which ends the proof. <)
8.4
Alternative proof of Desch's theorem
We present in this section an approach which is, in principle, different from the previous ones. The perturbed semigroup is constructed globally in time as the solution of a suitable Fredholm equation whose spectral radius is estimated. This proof was introduced in [7J (see also Chapter 12) in the context of scattering theory in L1 spaces but it has an independent interest. We are faced with solving the Duhamel equation
V(t)x = U(t)x + lot V(t - s)BU(s)xds which is clearly equivalent to
V(t)x = U(t)x + lot V(t - s) BU(s) xds where
= e-AtU(t)
U(t)
j
V(t)
= e-AtV(t)
and .>. is chosen such that
(8.7) We introduce the Banach space
H =
{t E [0, oo[ --+ L(L 1 (JL))j strongly continuous and uniformly bounded}
equipped with the norm
!!Z!!
= ~~~ IIZ(t) IIL(L' (I-'))
j
ZE H
and define the linear operator
L: Z The basic result is
E
H
--+
lot Z(t - s)BU(s)ds.
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Topics in Neutron Transport Theory
Lemma 8.4 L E L(H) and rcr(L) ~ rcr(B()" - T)-1).
Proof: Let x E D(T), x ;:::
LZ(t)x
o. Starting from =
lot Z(t - s)BU(s)xds
and using the additivity of the L1 norm, we derive the estimates
J t
~
ILZ(t)xl
IIZ(t -
s)II IBU(s)xl ds
o
J
J
00
< IIZII
00
IBU(s)xl ds = IIZII
BU(s)xds
o
0
IIZII IB().. - T)- 1x l ~ IIZII IIB().. - T)- 1 11I x l·
(8.8)
Now let x E D(T) be arbitrary and let x+, x- be its positive and negative parts. Then .l
xn±
= n Ion U(t)x±dt E D(T)
and
--t
x± in L 1(/1-)
.l
x n+ - xn_ = n
Ion U(t)xdt
--t X
in D(T) .
Hence t
ILZ(t)(xn+ - xn-)I
=
J
Z(t - s)BU(s)(xn+ - xn_)ds
o
< IIZII IIB().. - T)-111 Ilxn+1 + Ixn-IJ · (8.9) Passing to the limit,
ILZ(t)xl ~ IIZII IIB().. - T)-111Ixl
; x
E
D(T) .
(8.10)
Thus, LZ(t) E L(L1(/1-)) and
IILZII
= sup IILZ(t) II (2::0
~ IIZII IIB().. - T)-111·
(8.11)
193
Chapter 8. On positive co-semigroups in Ll (/-L) spaces We note that, for x E D(T), 1
t E [0, oo[ -+ LZ(t)x E L (/-L) is continuous.
Let x E Ll(/-L) and let Xn E D(T), Xn sup ILZ(t)xn - LZ(t)xml ~
-+
x, then (8.10) shows that
IIZIIIIB(A -
T)-llllxn - xml
(2::0
whence {LZ(t)xn;n E IN} is a Cauchy sequence in Cb([O,oo[;Ll(/-L)) and therefore LZ is strongly continuous. Finally L E L(H) and
IILII ~ IIB(A - T)-III· It follows, from (8.8), that
ILZ(t)xl ~ IIZIIIB(A - T)-lxl for x ::::: 0 and, for arbitrary x,
The estimate (8.12) shows, for Z fixed, that LZ(t) depends strongly continuously on
B : D(T)
-+
L 1 (/-L) endowed with the strong operator topology.
We approximate B by bounded operators as follows. Let
1 .1
R j : x E L 1 (/-L)
j
-+
j
U(s)xds E D(T).
Then
Rj : D(T) whence B j
= BRj
-+
D(T) goes to Id strongly as j
-+ 00
is a bounded positive operator and
BRj : D(T)
-+
L 1 (/-L) goes to B strongly as j
Let Lj be the operator corresponding to B j ,
LjZ(t) A calculation shows that
=
1t
Z(t - s)BjU(s)ds.
-+ 00 .
(8.13)
Topics in Neutron Transport Theory
194
Hence, for x 2: 0, ILjZ(t)xl
=
IIZIII/: dtn
I:n
= IIZIIII
2
tl)·· ·BjU(t -
dtn-l ... / : dtlBj U(t2 -
tn)xl
BjU(Ul)··· BjU(Un)xdf.Ll · ·· df.Lnl
Ul +· ·· Un~t
Letting j
~ 00
we get
ILn Z(t)xl
~ IIZIII (B()" -
T)-l) n xl
;n
E
N, x 2:
o.
Let x E L1(f.L) be arbitrary. Then ILnZ(t)xl
~
IIZII [I (B()" - T)-lr x+1 +
~
IIZIIII(B().. - T)-lrll [lx+1 + lx-I] = IIZIIII(B()" - T)-ltlllxl·
1
(B()" -
T)-lr x-IJ
It follows that IILnZ(t)1I
and
~ IIZIIII(B().. - T)-lrll ; t 2: 0
IILnl1 ~ II(B()"-T)-lrll ; n
E N
so that which ends the proof. <> Now the Duhamel equation reads
V=U+LV; VEH
(8.14)
Chapter 8. On positive co-semigroups in £l(p,) spaces
195
where U(t) = e-AtU(t) E H. According to (8.7), ru(L) < 1 and then (8.14) is solved by 00
V= LLnU o
(8.15)
where the series (8.15) converges in H (Le. uniformly in t ~ 0). This is nothing but the Dyson-Phillips expansion from {e-AtU(t); t ~ O}. It follows, as in Chapter 7, that V is a eo-semigroup with generator T )..J + B . Of course, if we multiply (8.15) by eAt, the series (converging uniformly in bounded times) is nothing but the Dyson-Phillips expansion from {U(t); t ~ O} . <:)
8.5
Comments
We point out that Theorem 8.1 was given first by W. Desch [5J . The proof of Desch's theorem via Miyadera perturbations (Theorem 8.2) and Lemma 8.1 are due to J. Voigt [9J while Theorem 8.3 is due to M. Mokhtar-Kharroubi [6J . We do not know whether Lemma 8.3 exists in the literature; the proof we give is due to M. Mokhtar-Kharroubi and W. Arendt (unpublished). The alternative proof of Desch's theorem in Section 8.4 is due to M. MokhtarKharroubi and is adapted from a similar theorem by M. Mokhtar-Kharroubi [7J. Miyadera perturbations are used by J. Voigt [8J to deal with substochastic semigroups in Ll spaces. We point out that the Ll techniques of this chapter extend to general Banach lattices for positive finite rank perturbations (see W . Arendt and A. Rhandi [lJ and M. Chabi and M. Mokhtar-Kharroubi [3]) .
References [lJ W. Arendt and A. Rhandi. Perturbations of positive semigroups. Arch. Math. 56 (1991) 107-119. [2J W . Arendt. Resolvent positive operators. Proc. Lond. Math. Soc. 54 (1987) 321-349. [3J M. Chabi and M. Mokhtar-Kharroubi. On perturbations of positive co-(semi)groups on Banach lattices and applications. J. Math. Anal. Appl. 202 (1996) 843-861. [4J F . Chatelin. Spectral Approximation of Linear Operators. Academic Press, 1983.
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[5J W. Desch. Perturbations of positive sernigroups in AL-spaces. Preprint, (1988) . [6J M. Mokhtar-Kharroubi. Characterisation BV des perturbations de Miyadera et applications. Workshop "Evolution equations, Control theory, Biomathematics". March 8-12, 1993. [7J M. Mokhtar-Kharroubi. Limiting absorption principles and wave operators on Ll(/-L) spaces. Applications to Transport theory. J. Funct. Anal. 115 NOl (1993) 119-145. [8J J. Voigt. On substochastic CO-sernigroups and their generators. Semes terbericht Funktionalanalysis, Thbingen, Wintersemester, (1984/85). [9J J. Voigt. On resolvent positive operators and positive CO-sernigroups in AL-spaces. Semigroup forum . 38 (1989) 263-266.
Chapter 9
On singular neutron transport equations in Ll spaces 9.1
Introduction
By singular transport equations we mean neutron transport equations involving singular cross-sections, that is unbounded collision frequencies and unbounded collision operators. Thus the standard perturbation theory fails. The basis of the following analysis is that unbounded and non-negative collision frequencies act as strong absorptions which allow for unbounded (to some extent) collision operators. We will show how the L1 techniques of Chapter 8 provide a convenient tool to build a reasonable theory for such equations. We point out however that their mathematical analysis is much more involved than in the classical setting of bounded perturbations. We are concerned with the well-posedness and the time asymptotic behaviour of transport equations of the form d'lj; dt 'lj;w-
where (x, v) E
-v a'lj; ax - a(v)'Ij;(x, v, t)
+
J'"
K(v, v )'Ij;(x, v , t)dv
Rn
=
0
j
'Ij;(x,v,O)
n x Rn, n c
= 'lj;o(x,v)
(9.1)
R n is a smooth open set and
r _ = {(x,v) E an x Rnj 197
v.n(x)
< O}
198
Topics in Neutron 'J}ansport Theory
an.
where n( x) is the outward normal at x E We assume there exist c > 0 and A c Rn a closed set with zero Lebesgue measure such that
0"(.) E
L~c(Rn
-A) and O"(v)
~
c a.e.
(9.2)
and
(9.3) where the scattering kernel K (. , .) is non-negative. The meaning of the first part of (9.2) is that the singularities of the collision frequency are included in a set A of zero Lebesgue measure outside which 0"( .) is locally bounded. We note that the scattering kernel K (., .) does not define a bounded operator in Ll(Rn; dV) . Thus, we cannot appeal to the standard perturbation theory to infer that the Cauchy problem (9.1) is well-posed in the sense of semigroup theory. Let us introduce the weighted space
(9.4) and the co-semigroup in Ll(n x Rn)
U(t) : f
E
Ll(n x Rn) ---.. U(t)f = e-tCT(v) f(x - tv, v)X(t < s(x, v)) (9.5)
with generator T T'Ij; {
= -v~ - O"(v)'Ij;; 'Ij; E D(T)
D(T)
=
{'Ij; E
L~(n x Rn);v~
E Ll(n x Rn),'Ij;lr_ =
(9.6)
o}
where s(x,v) = inf {s > O;x - sv ~ n}. A basic observation is the following smoothing effect of {U(t) ; t
~
O}
Proposition 9.1 For t > 0, U(t) maps U(n x Rn) continuously into the
subspace L~(n x Rn). Proof: By using the estimate ze- z
::;
e- 1 for z
~
0, one sees that
1
100(v)U(t)f(x, v)l::; et If(x - tv, v)X(t < s(x, v)) I and consequently
1 IIU(t)fIIL~::; et
IIfll£1 . <>
We note that the resolvent of T is given by r(s,V)
(A - T)-l f = Jo
e-().+CT(V»t f(x - tv, v)dt.
A second observation, already present in (9.6), is
Chapter 9. On singular transport equations in Ll spaces
199
Proposition 9.2 D(T) is continuously embbeded in L~(n x Rn).
Proof: By using the estimate, valid for ReA + c > 0,
10 I(A - T)-l f(x, v)1 dx ~ ReA ~ O'(v) 10 If(x, v)1 dx we obtain
II(A - T)-
1
fliLl" ~ vERn sup
O'(V)
R A+ () e 0' V
IlfliLl
which ends the proof. <> It is easy to see that Assumption (9.3) is equivalent to
(9.7) and that
Thus, the collision operator is bounded on a subspace between the whole space and the domain of T . We note that K is T-bounded
and
(9.8) Finally, we point out that if 0'(.) is unbounded then sup vERn
9.2
O'(v)
ReA + 0'( v)
= 1 regardless of A.
Generation results
We start with Theorem 9.1 Let 1IKIIL(L~(Rn);Ll(Rn )) turbation of {U(t) ; t 2: O}.
< 1. Then K is a Miyadera per-
Proof It is a direct consequence of Lemma 8.2. <> This first generation result relies only on the size of K. We give now another one (independent of the size of K) which relies on compactness arguments.
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Topics in Neutron Transport Theory
Theorem 9.2 Let K : L";(Rn)
L1(Rn) be compact (or at least domThen K is a Miyadera perturbation of
--+
inated by a compact operator). {U(t); t ~ O}.
The proof is based on the following
Proposition 9.3 If K : L~(Rn)
L1(Rn) is dominated by a compact
--+
operator then (9.9)
Proof. We may assume, without loss of generality, that K is compact. The estimate (9.8) shows that K().. - T)-l depends continuously on
uniformly on Re).. ~ c' (c' > c). Therefore it suffices to give a proof for a class of operators dense in L(L~(Rn); L1(Rn)). By approximating K in L(L~(Rn); L1(Rn)) by finite rank operators (and using the linearity) it suffices to prove that
IIK().. - T)-l H().. -
T)-ll1
--+
0 as Re)..
--+ 00
where K and H are one rank operators with kernels
K(v,v')
= f(v)g(v')
, H(v,v')
= h(v)k(v')
where f, hE L1(Rn) and ;, ~ E Loo(Rn) . In view of Proposition 9.2, we have to prove that IIK().. - T)-l HIIL(L~(nXRn);£l(nXRn))
--+
0 as Re)..
--+ 00 .
We factorize K()" - T)-l Has
K().. - T)-l H = F.N>. .k where
k : 'l/J F : 'l/J
E
E
N>.: 'l/J
L~(n x L1(n) E
Rn)
--+
J
'l/J(x, v)k(v)dv
E
L1(n)
Rn
--+
L1(n)
'l/J(x)f(v)
--+
J
E
L1(n x Rn)
n N>.(x,y)'l/J(y)dy E L1(n)
Chapter 9. On singular transport equations in L1 spaces
201
and
rOO
N>.(x , y) = Jo
(>.
e- +"
(=» t
x - y x- y x - y dt tg(-t-)h(-t-)x(lx-yl::::s(x ' lx_yl))t n
It suffices that
(9.10) It is easy to see that ( Ig(z)llh(z)1 IIN>.IIL(Ll(n»:::: JRn Re,X+O'(z)dz
(9.11)
and there exists c, independent of hand Re'x ~ c' > c, such that (9.12) The estimate (9.12) allows us to assume, without loss of generality, that h is compactly supported and its support is included in R!' - A where A is the set of singularities of the collision frequency 0'( .). The estimate (9.11) becomes Ig(z)llh(z)1 (9.13) IIN>.IIL(L'(n» :::: Re'x + O'(Z) dz
1
supp(h)
and the boundedness of 0'( .) on supp(h) implies (9.10) . <) Remark 9.1 We point out that, in general, IIK(,X - T)-111 does not go to zero as >. --+ 00. For instance, in the whole space, i.e. = R n , and for
n
>'+c>O
IIK(>. - T)-111 = sup >. vERn
where k(v)
= J K(v', v)dv' . By
assumption
k(V~ ) + 0' V
*1 is bounded. If *1 ~ a > 0
in an open subset of Rn on which 0'( .) is not bounded, then SUPvERn >.~~iv) ~ a regardless of >.. Proof of Theorem 9.2: In view of Proposition 9.3, r,,(K(>' - T)-l) < 1 for>. large enough. It follows, from Lemma 8.1, that (>. - T - K)-l exists and is positive for >. large enough and we conclude by Theorem 8.2. <) The calculations in the proof of Proposition 9.3 imply the following
Corollary 9.1 Let
n be bounded and let the collision operator
Topics in Neutron Transport Theory
202
be dominated by a compact operator. Then
is weakly compact. In addition, if K is compact on L~(Rn) and ifn convex then K(J... - T)-l K is compact on L~(n x Rn) . Proof We assume first that K is compact on L~(Rn). According to the arguments above, it suffices to prove the weak compactness of
where
NA(x, y) =
x- y x- y 10rOO e- (A+a (=)) tg(-t -)h(-t-)X(lx t
yl ::; s(x,
x- y
dt
Ix _ yl)) tn'
By decomposing 9 and h into positive and negative parts, we may assume that 9 and h are non-negative. We may also assume that h is compactly supported with support included in R n - A. Hence N A (x, y) is dominated by
where
because 0-(.) is bounded on the support of h. The convolution operator N A is compact on L1(n) because N A(') E L1(Rn) and 0. is bounded (see, for instance, [1] p. 74) and therefore NA is weakly compact by domination. This ends the proof when K is compact on L~(Rn) . If K is only dominated by a compact operator K on L~(Rn), then
is dominated by K(J... - T)-l K which is weakly compact, according to the previous argument, and we end the proof by a domination argument. If K is compact on L~(Rn) and 0. is convex, then
x-y x(lx - yl ::; s(x, -I-I)) x-y
=1
and N A = N A is compact. This ends the proof of the second part.
0
203
Cbapter 9. On singular transport equations in L1 spaces
Thus, under the assumptions of Corollary 9.1, K(A - T)-1 is power compact in £l(n x Rn) and therefore, denoting by s(T) the spectral bound ofT,
a(T + K) n {)..; ReA> s(T)} consists of, at most, isolated eigenvalues with finite algebraic multiplicities. Since this is not sufficient, a priori, to infer the spectral properties of the perturbed semigroup, a direct analysis is considered in the following section.
9.3
The essential type of the perturbed semigroup
According to the theory in Chapter 7 (Theorem 7.2), the essential type of {V(t); t ~ O} is less than or equal to that of {U(t); t ~ O} provided that some remainder term of the Dyson-Phillips expansion is weakly compact in £l(n x Rn). We will deal with the second order term R2(t) . Two major difficulties arise. First, in the theory of Chapter 7, the terms of the Dyson-Phillips expansion are not explicit on the whole space. Their definition on the whole space is obtained by an (abstract) extension argument. Secondly, those terms involve strong time integrals of operators which are not locally bounded in time. It is to overcome such difficulties that several technical preliminary results are needed. First of all, the smoothing effect of {U(t); t ~ O} (Proposition 9.1) is inherited by V(t) = et(T+K) in the following form. Proposition 9.4 For any a > 0 there exists a constant M such that
The proof is based on another technical result we prove first. Let a > 0 be fixed and let AO be large enough so that (9.15) Let E be the space of strongly measurable mappings
i.e. for any f E £l(n x Rn) t E [0, a[
-+
Z(t)f E L1(n x Rn) is measurable
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Topics in Neutron Transport Theory
and such that
endowed with the norm
IIZIIE =
sup 11/11:9
r IIZ(s)fll ds.
(9.16)
10
The finiteness of IIZIIE follows from Baire's theorem (see, for instance, [IJ p. 15). We define the operator
L: Z
E
E
---7
LZ(t)
=
1t
Z(s)KU(t - s)ds ; t
E
[O,aJ
where U(t) = e->'otU(t). Then Lemma 9.1 L E L(E) and Ta(L) ::; Ta(K(AO - T)-1) .
Proof: Let f
E
L1(n x Rn) be non-negative, then LZ(t)f
=
1t
Z(s)KU(t - s)fds.
Hence, thanks to the additivity L1 norm on the positive cone,
J:
IILZ(t)fll dt
: ; J: J: : ; J: J~ : ; J~ dt
ds
J: J: = J~ J:
IIZ(s)KU(t - s)fll ds = IIZ(s)KU(T)fll dT
dT IIZIIE IIKU(T)fll =
ds
dT
IIZ(s)KU(t - s)fll dt
IIZ(s)KU(T)fll ds
IIZIIE J~ IIKe->'oTU(T)fll dT
= IIZIIE IIJ~ Ke->,oTU(T)fdTII = IIZIIE IIK(Ao - T)-1 f II·
Chapter 9. On singular transport equations in Ll spaces If 1 E Ll(n x R"') is arbitrary then the above calculation shows that
J:
IILZ(t)III dt
:::;
IIZIIE [lIK(>.o -
T)-l
1+11 + IIK(>.o - T)-l 1-11]
:::; IIZIIE IIK(>.o - T)-lll [111+11 + 111-111 =
IIZII EIIK('\o - T)-lIIIlIII ·
Thus
and
On the other hand, Lm Z(t)I is equal to
and, for non-negative
J:
1,
IILm Z(t)III dt
Making the change of variables
tm - t m - 1 = U m t -tm = Um+l
205
206
Topics in Neutron Transport Theory
and using Fubini's theorem we get
I:
IILm Z(t)/11 dt
: ; I: IUi~O dUI
=
.IUi~O
IIZ(Ul)KU(U2) ·· · KU(u m+1)/11 dU2· · · du mdu m+1
dU2 ··· dumdum+1
I:
dU11IZ(UI)KU(U2)· ·· KU(u m+1)/11
:::; IIZIIE.IUi~O dU2 ·· · dumdum+l IIKU(U2) ··· KU(um+1)/11 = IIZIIE
IIIUi~O
dU2 ··· dU mdU m+1 KU (U2) ·· · KU(um+d/ll
= IIZII E I [K(>.o - T)-I]m III· When IE Ll([2
I:
X
Rn) is arbitrary we obtain, by decomposing it,
IILm Z(t)/11 dt
:::; IIZII E [II [K(>.o - T)-I]m 1+11 + I [K(>.o - T)-I]m I-II] :::; IIZII EI [K(>.o - T)-I]mll [111+11 + III-Ill =
IIZIIE I [K(>.o - T)-I]mllll/ll ·
Hence
and
Thus (9.17)
which proves the claim.
<>
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Chapter 9. On singular transport equations in L1 spaces
Proof of Proposition 9.4 : We observe that
=
Ja J o
le-t(>'o+a(v)) f(x - tv, v)X(t < s(x, v))1 a(v)dxdv
dt
OxRn
:; 1 1 00
le-t(>'o+a(v» f(y, v)1 a(v)dydv ,
dt
.
0
whence
.
OxRn
JIl aU(t)fll £'
dt::; sup
vERn
o
tt)
a v + A0 Ilfll£' ,
(9.18)
i.e.
aU E E where a is the multiplication operator by a(.). According to the Duhamel equation in Chapter 7,
V(t) = U(t) + lot V(s)KU(t - s)ds
(9.19)
where V(t) = e-.>.otV(t). Then Z(t) = aV(t) satisfies the Fredholm equation
Z
= aU +LZ.
One sees, in view of (9.15) and (9.17), that
Z = (I - L)-lU E E which proves our claim since V(t) = e>.otV(t). <> Thanks to the smoothing effects of {V(t); t 2 O} one easily derives the following results whose proofs are left to the reader Proposition 9.5 The following Duhamel equation holds
V(t)f
= U(t)f + lot U(t - s)KV(s)fds ; f
E L1(0
X
Rn)
(9.20)
and the second order term of the Dyson-Phillips expansion is given by
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208
For the sequel we need the following stability result. Proposition 9.6 There exists c(t) locally bounded in t , independent of K,
such that
Proof. According to (9.21) , t
II R2(t)fll
::; Cl(t)
JJ 8
ds
o
0
JJ t
=
Cl(t)
t
dr
o
ds IIKU(s - r)KV(r)fll
r
J t
< Cl(t)
dr IIKU(s - r)KV(r)fll
t
dr / dp IIKU(p)KV(r)fll
o
(9.23)
0
where Cl(t) = sUPo<.,.
J:
dp IIKU(p)KV(r)fll
::;
IIKIIL(L~(Rn);£1(Rn))
J:
dp
IIU(p)KV(r)fIIL~
::; C2(t) IIKIIL(L~(Rn);£l(Rn» IIKV(r)fll£1 . Hence
Finally, by Proposition 9.4, there exists C3(t) such that
which ends the proof. (; We assume, from now on, that
Chapter 9. On singular transport equations in L1 spaces
209
To prove the weak compactness of R2(t) it suffices to restrict ourselves, thanks to Proposition 9.6, to finite rank collision operators. By linearity we are led to consider R2(t) =
fot ds fos drU(t -
s)KU(s - r)HV(r)
where K, H are one rank collision operators with kernels K(v, v) = Jr(V)g1 (v) ,H(v, v) = h(V)g2(V)
where Ii E L1(Rn) ,~ E Loo(Rn) (i = 1,2). The last procedure is to approximate R2 (t) by m(t) =
it
ds
fos-e drU(t -
s)KU(s - r)HV(r).
(9.24)
Unfortunately R2(t) does not, a priori, converge to R2(t) in the uniform topology because the norm of the integrand is not integrable. This is due to the unboundedness of the collision frequency a( .). To overcome this difficulty, we observe that R2(t) depends continuously on h E L1(Rn). Therefore there is no loss of generality in assuming that h is compactly supported and its support is included in Rn - A where A is the set of singularities of the collision frequency a(.). We have Proposition 9.7 Let h be compactly supported with support included in Rn - A. Then R2(t) -+ R2(t) in the uniform topology as c -+ O.
Proof We note that
where
JIe = J: J:-e ds
1 J: J: Je
IIIefl1 :::;
=
it
ds
ds
drU(t - s)KU(s - r)HV(r)
drU(t - s)KU(s - r)HV(r).
l~e dr IIU(t - s)KU(s -
r)HV(r)fll·
(9 .25)
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210
We first show there exists a constant M such that for t
IIU(t - s)KU(s - r)Hcpll£1 ::; M
~
~
s
r
IlcpllLlu ; cp E L~(n x Rn) .
(9.26)
We note that
KU(s - r)Hcp =
h(v) I gl(v")m(s - r, x, v")h(v")dv" I g2(V')CP(x - (s - r)v", v')dv'
where m(t,x,v) = e- t<1(v)X(t
< s(x,v)). Hence
IIU(t - s)KU(s - r)Hcpll£1 ::; IIKU(s - r)Hcpll£1 l
l
l
::; Ilhll£1 I
I Ilgl(v )m(s- r,x ,v )h(v )g2(v')cp(x-(s-r)v",v')1
::; Ilhll£1 I
I I Igl(v")h(v")g2(v')cp(y,v')1 dydv" dv'
::; Ilhll£1.f
In. 1•• "11 ~
II
"
II
<1(V") Ih(v)1 a(v )dv I I
I92(V ') I <1(V')
Icp(y, v
,
)1 a(v
::; Iihll£1I1.;;.llu" 111211£1 IlaliL'''>(supp(h» II~"L'''> IlcpliL~ which proves (9.26) . Thanks to (9.26) , we have
Illgfll ::; M
I: ds I:-g dr
IIV(r)fIIL~
= M It It X(E , t)(s)X(s - c:, s)(r) IIV(r)fll£1 drds o
whence
0
u
Illgli ::; c:MsuPllfIlL,9 J~ IIV(r)fIIL~ dr.
We point out that sup
{t IiV(r)fllL' dr < 00
IIfllLl9 Jo
u
,
)dydv
,
211
Chapter 9. On singular transport equations in L1 spaces
in view of Baire's theorem (see (9.16) and the proof of Proposition 9.4). Finally 111,,11 - t 0 as c - t o. We prove, by similar arguments, that 111,,11 - t 0 as c - t O. <> We are now ready to prove the main result of this section.
Theorem 9.3 We assume that the collision operator
is dominated by a compact operator and that
n is bounded.
Then R2(t) is
weakly compact in L1(n x Rn) . Proof We first assume that
is compact. In view of the preliminary results, it suffices that m(t) be weakly compact in L1(n x Rn). Let 'I/J E L~(n x Rn), then
U(t - s)KU(s - r)H'I/J =
r
k(x, v, r, s, x', V')'I/J(X/, v')dx' dv '
inxRn
where k(x,v,r,s,x/,v ' ) is equal to
m(t - s,x,v) h(V)gl(X - x' - (t - s)v) (s-r)n s-r xm ( s-r,x-(t-s)v, and m(t,x,v)
X -
x' - (t - s)V) (X - x' - (t - s)V) I f2 g2(V) s-r s-r
= e-tcr(v)X(t < s(x,v».
We factorize m(t) as
m(t) = Q2Q1 with
Q1
and
( )= J
Q2'I/J x,v
S,xnXRn
(
I
')
I
I
I
I
k x,v,r,s,x,v 'I/J(r,s,x,v )drdsdx dv.
Topics in Neutron Transport Theory
212
It suffices to prove the weak compactness of Q2' We note that k(x, v, r, s, x', v') is equal to 92(V') times
[m(t - s,x,v) h(v)h(x - x' - (t - s)v)m(s-r,x-(t-s)v, x - x' - (t - s)v)] (s-r)n s-r s-r where h = 9112 E
L1 (Rn).
Hence
u
where G: 'Ij; E
L~(SE: x n x
Rn) -; G'Ij;
=
r 'Ij;(r,s,x',V')92(V')dv'
JRn
E Ll(SE: x
n)
and
Q3'1j;(X,V) =
f
k(x,v,r,s,x',v')'Ij;(r,s,x')drdsdx'
. s.xn
where
-
, , k(x,v,r,s,x,v)
m(t - s,x,v) h(v)h(x - x' - (t - s)v) (s-r)n s-r xm(s-r,x-(t-s)v,
x-x' -(t-s)v ). s-r
It is not difficult to see that Q3 depends continuously on
(!I, h) E L1(Rn) x L~(Rn) . Therefore we may assume, without loss of generality, that hand hare continuous with compact supports and this implies the weak compactness of Q3, ending thus the proof when K is compact. If K is only dominated by a compact collision operator K, then the second order term of the DysonPhillips expansion corresponding to K is dominated by that corresponding to K and we conclude by a domination argument. 0 Corollary 9.2 We assume that the collision operator
K : L~(Rn) -; Ll(Rn)
is non-negative and dominated by a compact operator. Then A = T +K with domain D(A) = D(T) generates a co-semigroup {V(t); t ~ O} on £1(n x Rn). Moreover, if n is bounded then the essential type of {V(t); t ~ O} is less than or equal to that of {U(t); t ~ O} . Proof: It follows from Theorems 9.2, 9.3 and 7.2.
0
213
Chapter 9. On singular transport equations in L1 spaces
9.4
Comments
The material of this chapter, due to M. Chabi and M. Mokhtar-Kharroubi, is a part of M. Chabi's thesis [2] . Such singular transport equations are motivated by cross-sections of free gas models studied by A. Suhadolc [5] in L1 spaces. However, A. Suhadolc worked in the space L1(n x Rn; dxM(v)dv), instead of L1(n x Rn), where M(v) is a suitable Maxwellian in order to recover the boundedness of the collision operator and to keep the usual framework of bounded perturbation theory. The generation result corresponding to the case J K(v',v)dv' :::; a(v) (Theorem 9.1) is proved by J. Voigt [6] as a consequence of abstract results on substochastic semigroups which are themselves related to Miyadera perturbations. We point out that Theorem 9.1 covers only the case IIKIIL(L~(Rn);L'(Rn)) < 1. However, it is possible to deal with the case IIKIIL(L~(Rn) ; L'(Rn)) = 1 by a limiting process (see [6]). We point out also that nonhomogeneous collision frequencies and scattering kernels can also be dealt with by similar techniques. The spectral theory of the perturbed semigroup (Corollary 9.2) relies essentially on the assumption that K : L~(Rn) - t £1(Rn) be dominated by a compact operator. It is an open question whether this hypothesis is satisfied by the physical model treated by A. Suhadolc [5] . We point out however that such compactness assumptions of the collision operator with respect to velocities are almost necessary in the context of bounded perturbations (see Chapter 4). Two generation results are given in this chapter. The first one (Theorem 9.1) relies on the assumption IIKIIL(L~(Rn);L'(Rn)) < 1 while the second one (Theorem 9.2) is based on the compactness of K : L~(Rn) - t L1(Rn) regardless of its size. Actually, if K = K1 +K2 where K1 : L~(Rn) - t L1(Rn) is dominated by a compact operator and if IIK21IL(L~(Rn);L'(Rn)) is small enough then T + K is a generator. Indeed,
IIK
T)-lll
is small where r a (K1().-T)-1) < 1 for)' large enough and 2 (). regardless of >. , so that r a (K (>. - T) -1) < 1 for >. large enough because of the upper semicontinuity (with respect to the operator norm) of the ~pectral .radius ([3] p. 20~). W~ conjecture that IIK21IL(L~(Rn);L'(Rn)) < 1 IS suffic2ent. The analysIs of smgular neutron transport equations in £P spaces (1 < p < 00) is dealt with by M. Mokhtar-Kharroubi and J. Voigt
[4].
214
Topics in Neutron Transport Theory
References [1] H. Brezis. Analyse Fonctionnelle: Theorie et Applications. Masson, Paris, 1983. [2] M. Chabi. TMorie de scattering dans les espaces de Banach reticules. Transport singulier dans £1. These de l'Universite de Franche-Comte, 1995. [3] T . Kato. Perturbation Theory for Linear Operators. Springer Verlag, 1984. [4] M. Mokhtar-Kharroubi and J. Voigt. On singular neutron transport equations in V spaces. Work in preparation. [5] A. Suhadolc. Linearized Boltzmann equation in L1 space. J. Math. Anal. Appl. 35 (1971) 1-13. [6] J . Voigt. On substochastic eo-semigroups and their generators. Semesterbericht Funktionalanalysis, Thbingen, Wintersemester, (1984/85), 115.
Chapter 10
Stochastic formulations of neutron transport. Nonlinear problems 10.1
Introduction
This chapter deals with a class of nonlinear neutron transport problems arising in the probabilistic approach of neutron chain fissions . Neutron transport theory deals ordinarily with expected value of neutron populations . In order to describe the fluctuations from the mean value of neutron distributions , stochastic formulations of neutron chain fissions have been introduced very early, in particular by L. Pal [11], G.I. Bell [3] [1] and M. Otsuka and K. Saito [10] . Two main mathematical problems, related to such formulations, are considered in this chapter. We first study the stationary equation
8r.p v.--- -a(x, v)r.p(x, v) 8x -a(x, v) x
[1 - co(x, v) - f
J
Ck( X,
v,
v~ , ... , v~)
k=1 Vk , " x (l- r.p ( x , VI) · ·· (1- r.p(x, vk)dv i
. ..
I dVk))
1
(10.1)
with the conditions (10.2) 215
216 where m
Topics in Neutron Transport Theory ~
En x V and r + = {(x, v) E an x V; v.n(x) > O}.
2 is an integer, (x, v)
We use the notation Vk = V X . . . X V (k times) and assume that the velocity space V is the unit ball of R':; with normalized Lebesgue measure. Next, we will deal with the evolution problem
a'l/J at - v. a'l/J ax + a(x, v)'l/J(t, x, v)
a(x,v) x [l-CO(X,V) -
f. JCk(X,V,V~,
...
,v~)
k=l Vk
x(l-
'l/J(t,x,v~) ... (1- 'l/J(t,x,v~)dv~ ... dV~))l
(10.3)
with the conditions
'l/J(O, x , v) = 'l/Jo(x, v) , 'l/J(t,., .)Ir + = 0 , 0::; 'l/J ::; 1.
(lOA)
We also study the time asymptotic behaviour of its solutions. Let us recall briefly the physical meaning of the equations above and refer to G.I. Bell [3] [1] for details. In a multiplying medium occupying a region ncR';, a neutron, interacting with the nucleus of the host material, may be absorbed, scattered in random directions or may produce (instantaneously), by fission process, more than one neutron. The probability that a neutron, with velocity v E V, at position x E n, yields, by fission process, i neutrons (1 ::; i ::; m) with velocities v~, ... , v: is denoted by
and
(10.5) where eo(x, v) is the probability to be absorbed. An important information is provided by the probability pj(tf, x, v, t) j
= 0, I, ...
that a neutron, born at time t with velocity v and position x, gives rise to > t. The probabilities Pj(t f, x, v, t) (j = 0, 1, ... ) are
j neutrons at time t f
217
Chapter 10. Nonlinear problems
governed by infinitely many coupled equations [15] [16]. On the other hand, the probability generating function 00
G(z,x,v,t,tf):= L,Zipi(tf,X,v,t); t
< tf
o is governed by a nonlinear backward equation (see [3]) with the final condition G(z,x,v,tf,tf) = z and the boundary condition G(z,x,v,t,tf) = 1; (x,v) E r+, t
< tf .
This chapter is devoted to the mathematical analysis of the probability generating function. Mathematically speaking it is expedient to consider 7/J(z, x , v, t) := 1 - G(z, x , v, tf - t, tf)
which is governed by the initial boundary value problem (10.3) with initial condition 7/Jo(x, v) = 1 - z and homogeneous boundary condition. Equation (10.1) governs the probability of a divergent chain reaction. We note, in view of (10.5), that the stationary problem (10.1) (10.2) admits
10.2
Preliminary results
We will assume throughout that we assume that
nc
Rn is bounded and convex. Moreover,
Ci( .) are non-negative and bounded (1 :::; i :::; m)
Topics in Neutron Transport Theory
218 and that
= a(v)
a(x,v)
, infa(.)
= >.* > O.
(10.6)
We formulate (10.1) (10.2) as a fixed point problem for a suitable operator and derive some preliminary properties. We define the following operator a'IjJ
= v. ax
T'IjJ
- a(x, v)'IjJ(x, v) ; 'IjJ E D(T)
with domain
= {'IjJ E LOO(n x V);
D(T)
v. ~~ E LOO(n x V), 'ljJlr +
= O} .
In view of (10.6), (0 - T)-1 exists and
11(0 - T)-11IL(L<Xl(fl XV))
~
;*.
We note, in view of (10.5), that the right-hand side of (10.1) is equal to
-a (v)
t iv(k
Ck(X,
v,
k=1
v~, .. ., v~) [1 - IT (1- 'IjJ(x, V~))l'
(10.7)
3=1
Thus, we can write (10.1) in the form -T'IjJ = a(v)
t k=1
(k
iv
Ck(X,
v,
v~, ... , v~) [1- IT (1 -'IjJ(x, V~))l dv~ ... dv~ 3=1
~
with the condition 0
'IjJ
~
1. Introduce the set
B = {'IjJ E LOO(n x V); 0
~
'IjJ
~
I}
and the nonlinear operator in LOO(n x V) N: 'IjJ
--t
a(v)
t iv(
k
k=1
Ck(X,
v,
v~, ... , v~) [1 - IT (1 - 'IjJ(x, V~))l dv~ ... dv~. 3=1
We may formulate (10.1) (10.2) in operator form
or
(10.8)
where
N=(O-T)-IN. This section is devoted to some basic properties of N and to related results.
219
Chapter 10. Nonlinear problems
LelllIIla 10.1 The operator N leaves B invariant and is nondecreasing.
Proof Let 'ljJ E B, i.e. 0::; 'ljJ(x, v) ::; 1. Then, for alII::; k::; m, k
0::; 1-
II(l- 'ljJ(x,v~))::; 1 on 0 x Vk j=1
(10.9)
On the other hand, r(x,v)
(0 - T)-1cp(X, v) = Jo
e-a(v)scp(x + sv, v)ds
where
s(x,v)
= inf {s > O;x + sv
Thus
r(x,v)
(0 - T)-1a-::; a-(v) Jo
e-a(v)s ::; 1
and, using (10.9) ,
N'ljJ = (0 - T)-1 N'ljJ ::; 1 which shows the first claim. Consider the function k
Fk : (Z1' ... , Zk) E [0, l]k
-+
II (1- Zj).
1-
(10.10)
j=1 It is clearly nondecreasing on [0, 1] k. This shows that N and hence also nondecreasing in B . <)
N are
To proceed further , we introduce some relevant operators. Let K be the linear operator on Loo(O x V)
cp-+a-(v)f fk k=1 Jv
Ck(X, V,V~, ... ,v~) [tCP(X'V~)j dv~ ... dv~ 3=1
(10.11)
and let L be the nonlinear operator defined by
Up
~ a(v) t, Lc,(x, v , v;"
v;)
[ll
(1 -
~(x, v;)) - 1+ t, ~(x, v;) j.
We note that N = K-L.
(10.12)
220
Topics in Neutron Transport Theory
Lemma 10.2 The operator L is non-negative. Proof We consider the function k
F k : (Zl' ... , Zk) E
[0, IJk
--.
II (1 -
k
Zj) - 1
+L
j=l
Zj.
(10.13)
j=l
We note that
and that
(10.14) Thus Fk cannot reach its minimum in JO, l[k. To show that Fk is nonnegative, it suffices to show that it is non-negative on the boundary of k [O,IJ . Let us show this by induction. We note that F1(Z) = 0. We assume k that Fk-1(Z) ;::: and we choose Z E 8 [O,IJ ,i.e.
°
:3 1 ::; i ::; k ; Zi = 0 or Zi = 1. If Zi
= 1, then k
Fk(Z) =
L
Zj;::: O.
j=l.#i
If Zi = 0, then k
L j=l.#i
Zj = Fk-1(Z)
j=l,j¥i
°
where z = (Zl, .. . , Zi-1, Zi+l, ... , Zk). Hence Fk(Z) ;::: from the induction hypothesis. Thus, for each
II (1- 'IjJ(x, v~)) j=l
k
1+
L
k
,
(1::; k ::; m)
j=l
which proves the claim. 0 We give now a compactness result. Lemma 10.3 The operator [(0 - T)-lK]2 is compact in LOO(n x V).
221
Chapter 10. Nonlinear problems Proof. We define, for 1 ::; j ::; k, the operator
Cl:
-+
a(v)
i
ck(x,v,v')
where c}Cx,v,v') = C1(X,V,V') and, for k:::: 2,
Then (10.11) amounts to m
k
K=LLCl k=l j=l
and consequently m
k
(0 - T)-l K = L L(O - T)-lC{ k=l j=l
We note that [(0 - T)-l K] 2 involves operators of the form
(0 - T)-lCl(O - T)-lC{ , and Cl(O - T)-lC~, are compact in LOO(D. x V), by duality from the L1 compactness result in Theorem 4.4. <; We end this section with a result on smoothing effects of some nonlinear operator. We decompose the polynomial operator N as m
where N1
=K
is the linear part, N2 is the quadratic part of the form
N2
=
and, more generally,
where
r P2(X, v, iV2
V1,
V2)
222
Topics in Neutron Transport Theory
Lemma 10.4 Let r
> nm. Then the nonlinear operator
N(O - T)-1 N ; LOO(0. x V)
----t
LOO(0. x V)
extends as a continuous operator from Lr(0. x V) into LOO(0. x V) . Proof We note that m
N(O - T)-1 N
= L Ni(O - T)-1 N i=1
and that Ni is monomial of degree i. Let
'l/Jj
= (0 - T)-1 Nj
then Ni(O - T)-1 N
t .. t iv'{.
Pi(X, v, WI, ..., Wi)'l/Jj, (x, wt}··· 'l/Jj; (x, Wk) .
j,=1
j;=1
It suffices to restrict ourselves to one term of the sum above. To avoid tedious calculations we consider (for simplicity only) the case where jl = ... = ji = j. Thus we deal with
f
Pi(X,W,Wl, .. . ,Wi)(O-T)-INj
v; (10.15)
Actually, this term is nothing but Ni(O - T)-1 Nj
f
S(X'W)
o
e-a(w)sds
f
. Pj(x - SW, W, VI, ... , Vj)
vJ
x
223
Chapter 10. Nonlinear problems
Making the change of variables x~ = x - SWe (1 ::; e ::; i) dx: = sndw e , the absolute value of this integral is dominated by a constant times (10.16) where
Also (10.16) is dominated by a constant times
IT' 1I [~(x~)r 'I
e=l
!1 X -
Xe
n- 1
dx' e
=
[1 I [~(Y)ljI !1 X -
Y
n- 1
d
l'
Y
By a simple convolution argument, one sees that, for p > n,
Hence, for r > nm,
The claim follows from the fact that £00(0. x V) is dense in £r(0. x V) . 0
10.3
Maximal solution of the stationary problem
We are ready to study the fixed point problem (10.8). We first show the existence of a maximal solution. Theorem 10.1 Problem (10.8) has a maximal solution.
Proof We define inductively a sequence {.,pk}
According to Lemma 10.1,
Topics in Neutron Transport Theory
224
By using Lemma 10.1 repeatedly, it follows, by induction, that
0:::; 'l/Jk+l Thus {'l/Jk} such that
c B and is 'l/Jk
In particular, 'l/Jk
--+
--+
= N'l/Jk :::; N'l/Jk-l = 'l/Jk :::; 1.
(10.17)
nonincreasing. It follows that there exists V5 E B
V5 pointwise and monotonically.
V5 in Lr(rl x V) for all finite r. Let us show that 'l/Jk
--+
V5 in LOO(rl x V) .
(10.18)
Since
'l/Jk+l = (0 - T)-l N'l/Jk , it suffices to show that the sequence {N'l/Jd converges in LOO(rl x V) . But
N'l/Jk+l = N(O - T)-l N'l/Jk so (10.18) follows from Lemma 10.4. Thus (10.19) We mention that (10.18) is not necessary to deduce (10.19) which can be obtained by passing to the limit by the monotone convergence theorem. However (10.181.. will playa basic roleJn Section 10.9. Let I.{J E B be another fixed point of N . Then, from I.{J = NI.{J :::; 1 = 'l/Jo , one sees, by induction, that I.{J:::; 'l/Jk V k and then I.{J :::; V5 thus showing that V5 is the maximal solution. <> We remark, at this stage, that we have no guarantee that V5 is not the zero (trivial) solution of (10.8) . The existence of a nontrivial solution turns out to be connected to spectral properties of the linearized problem. To this end we give some definitions. We mention, in view of the power compactness of (0 - T)-l K, that
a(T + K) n {..\; Re..\ > -lim inf a(v)} v-+o
consists, at most , of isolated eigenvalues with finite multiplicities. Moreover, if this point spectrum is not empty then there exists a leading eigenvalue. The latter result can be proved directly or derived by duality from the corresponding result in Ll (0, x V) which follows from general results on positive co-semigroups (Theorem 5.2) . We say that Problem (10.8) is sub critical (resp. critical, resp. super critical) if the spectral bound
s(T)
:=
sup {Re..\; ..\
E
a(T + KH
225
Chapter 10. Nonlinear problems
is negative (resp. equal to zero, resp. positive). It is easy to see that this amounts to T(1 [(O-T)-lK] < 1 (resp. T(1 [(O-T)-lK] = 1, resp. T(1 [(0 - T)-1 K] > 1). We first give nonexistence results. We start with the easiest one.
10.4
The subcritical case
Theorem 10.2 If T(1 [(0 - T)-l K] < 1 then (10.8) has no nontrivial solution.
PToof Let
ep=Nep; epEB , ep=rfO, i.e. Using the decomposition N = K - L and the fact that L is non-negative,
(O-T)-lKep?,ep; ep?,O, ep=rfO which implies that
10.5
T(1
[(0 - T)-l K] ?, 1.
0
The critical case
We give here another nonexistence result which is more involved than the one above. We introduce the assumption I
:3 2::S: ko ::s: m;
I
Ck o (X , V,v 1 , ... ,Vko)
> 0 on
n x V k +1 0
•
(10.20)
Then
Theorem 10.3 Let T [(0 - T)-l K] = 1. If (10.20) is satisfied then (10 .8) has no nontrivial solution. (1
Proof Assume there exists a nontrivial solution '1jJ '1jJ ?, 0 , '1jJ =rf 0 on
n x v.
We note that ko
ko
j=l
j=l
II (1 - '1jJ(x, v~)) -1 + .L ep(x , v~) ?, 0 on n x Vk
Topics in Neutron Transport Theory
226
and Fko(z), defined by (10.13), vanishes only ifzi = 0 for alli E [1,2, ... , koJ. It follows, from (10.20), that
is strictly positive and consequently L'lj; > 0 a.e. on N'lj; < K'lj; a.e. on
n x V. Hence
n xV.
(10.21)
Let To and Ko be the operators in Ll(nxV) such that TO' = T and KG = K . Since Ko(O-To)-1 is power compact in Ll(n x V), and irreducible because [Ko(O - TO)-1]2 is positivity improving in view of (10.20), then there exists a strictly positive 'lj;o E Ll(n x V) such that
because ru
[Ko(O - TO)-I] = ra [(0 - T)-1 K] .
Hence, using (10.21) ,
where (., .) is the pairing between Ll(n x V) and Loo(n x V). This ends the proof. <;
10.6
The snpercritical case
We begin with an existence result Theorem 10.4 If ru [(0 - T)-1 K]
> 1 then (10.8) has at least one non-
trivial solution. Proof. We recall that k
Fk : (Zb ..., Zk)
E
[0, IJk
-t
1-
IT (1 -
Zj)
j=1 and note that
Fk(O) = 0 ,
~~: (0) =
1 (1 SiS k).
(10.22)
227
Chapter 10. Nonlinear problems Thus there exists ~ E [0, Ilk, 0 ~ ~i ~ Zi such that k
8F k
H(z) = LZi 8z. (~). i=l
(10.23)
•
On the other hand, for each 0 < 1-£ < 1, there exists co
> 0 such that
so that, in view of (10.22) ,
8Fk -(z) 8zi
~
8Fk (1- 1-£)-(0) . 8zi
and, in view of (10.22) (10.23),
. k 8Fk k Fk(Z) ~ (1- 1-£) LZi 8z (0) = (1- 1-£) LZi for 0 ~ Zi ~ co (1 ~ i ~ k), i=l'
i=l
i.e.
1-
k
k
j=l
i=l
II(I- Zj) ~ (1- 1-£) LZi
for 0 ~ Zi ~ co (1 ~ i ~ k) .
(10.24)
Let 'ljJ* be a non-negative eigenfunction of (0 - T)-l K corresponding to its spectral radius and such that 11'ljJ*lluX>C!1xv)
= 1.
We define cp = c'ljJ* with c ~ co.
lt follows, from (10.24), that
1and then
k
k
j=l
j=l
II (1- cp(x, v~)) ~ (1 - 1-£) L cp(x, v~) , (1 ~ i ~ k)
(10.25)
Topics in Neutron Transport Theory
228 Hence
Nrp? (1- J-L)Krp and
Nrp? (1- J-L)(O - T)-1 Krp = (1 - J-L)ra [(0 - T)-1 K] rp.
(10.26)
We choose J-L in such a way that (10.27) Therefore Nrp ? rp, i.e. rp is a subsolution. We define inductively a sequence {rp d by rpo = rp and rp k+1 = Nrp k . By arguing as in the proof of Theorem 10.1, one verifies that {rpd C B is nondecreasing and converges in LOO(0. x V) to '1/;, a nontrivial fixed point of N. <> To prove that this solution is the minimal solution we need a technical result. Lemma 10.5 The solution given by Theorem 10.4 is independent of c. Proof Let 0 < Cl < C2 < co and let '1/;1, '1/;2 be the corresponding solutions. We set rpl = Cl '1/;* , rp2 = c2'1/;*·
From the order relation rpl :S rp2 and the construction of the solutions, it follows that '1/;1 :S '1/;2. Assume momentarily that (10.28) then '1/;1 is an upper bound of the inductive sequence which gives the solution '1/;2 and consequently '1/;2 :S '1/;1 and this implies the equality of the solutions. Thus it suffices to prove (10.28) . Define the set
E = {c; 0 < c :S C2, c'l/;* :S 'l/;1} . We note that E is a closed and nonempty interval (cl E E). Let c* be the least upper bound of E. Assume that c* < C2. By using (10.26),
(1 - J-L)ra [(0 - T)-1 K] x (c*'I/;*)
= (1 - J-L)(O - T)-1 K(c*'I/;*)
:S N(c*'I/;*) :S N('I/;I) = '1/;1' Therefore
c*(l-J-L)ra [{O-T)-IK] '1/;* :S'I/;1 which contradicts the definition of c· in view of (10.27). Thus c* hence (10.28) follows. <> We are ready to prove
= C2
and
229
Chapter 10. Nonlinear problems
Theorem 10.5 We assume that ru [(0 - T)-1 K] > 1 and that
:1(~):=.inf(X'VI)EnX vCl(X'V'~/»O {
onV
(10.29)
c(v):= inf(x,v)En x vCl(X,V,V) > 0 on V.
Then (10.8) has a nontrivial minimal solution. Proof Let us show that the nontrivial solution 'IjJ given by Theorem 10.4 is the minimal solution. Let cp be another nontrivial solution. To show that 'IjJ :S cp it suffices, according to lemma 10.5, to prove that
:3 c > 0 such that c'IjJ* :S cp
(10.30)
since cp would be an upper bound of the inductive sequence starting at c'IjJ* and tending to 'IjJ. We set Q
:= ru [(0 - T)-1
K].
Since and (10.31) it suffices to prove that
which is nothing but
In view of the positiveness of (0 - T)-I, it suffices that
Finally, this holds for c small enough if there exists c > 0 such that
Ncp 2: c.
(10.32)
We note that NCP=a(v)tlk k=1 v
Ck(X,V,V~ , ... ,v~) [1- J=1 rr(1-cp(X'V~))l dv~ . .. dv~.
Topics in Neutron Transport Theory
230 Hence Nt.p ~ a(v)
, " 1 dv l = C 1 t.p Jv( C1(X,V,V1)t.p(X,V1)
and, in view of (10.31) ,
so that (10.33) On the other hand, C 11 (0 - T) -1 C 11 t.p =
1
"
"
H(x, x , v, v )t.p(x , v )dx dv J
I
f!xV
where H(x, x', v, v') is equal to
1
00
o
and
x-x' x - x' x - x' ' x - x' ,ds e- a (-,-)sa(v)c1(X, v, --)a(--)c1(X , - - , v )~
s
C1 (., ., . ) I
s
s
is extended by zero outside V. It follows that
roo e-a(-s-)Sh(-s-) x - x' ds sn = G(x -
2
I
x-x'
H(x, x, v, v ) ~),* Jo
where h(v)
s
= £:1 (v)~(v). cf(o -
, x)
>0
Thus
T)-lCft.p
~
in
G(x - x')<j)(x')dx' := ~(x)
Iv
where <j)(x') = t.p(x', v')dv'. We note that <j) f=. 0 since t.p f=. 0, and that ~(x) > O. Finally, a simple convolution argument shows that ~(.) is continuous on R n whence ~( . ) is bounded away from zero on 0 and therefore (10.32) is satisfied, in view of (10.33) . <>
10.7
On the uniqueness
This section is devoted to the uniqueness of nontrivial solutions. Uniqueness turns out to be tied to a geometric property of certain operators related to N. We start with a useful definition. Definition 10.1 A non-negative nonlinear operator A defined on B c LOO(O x V) is said to be I-concave if, for any '1jJ E B, '1jJ f=. 0 and 0 < t < 1, there exist a, {3, J..L > 0 such that A(t'1jJ) ~ (1 {
a
+ J..L)tA('1jJ)
'5: A('1jJ) '5: {3.
231
Chapter 10. Nonlinear problems
We introduce the nonlinear operators Ck (1 ~ k ~ m) on LOO(0. x V) Ck 'ljJ
r
=
IT(V)Ck(X, v ,
iV k
v~, ... , v~) [1 - IT (1 - 'ljJ(x , V~))l dv~ . . . dv~ j=l
and give a preliminary uniqueness result. Theorem 10.6 If the operators CkN (1 ~ k ~ m) are I-concave, then (10 .8) has at most one nontrivial solution.
(3L
J.L1) Proof Let'ljJ1 and 'ljJ2 be two nontrivial solutions of (10.8). Let (01, the I-concavity of CkN and
= 1, 2) be the parameters corresponding to 'ljJi (i = 1,2) . We have (i
-
1
Ok
2
Ok
-
Ok
Ck'ljJ1 = Ck N 'ljJ1 2: Ok = {3~{3k 2: {3~CkN'ljJ2 = {3~Ck'ljJ2
which shows that the set
is not empty. Let tk := sup {t
> 0; C k'ljJ1 2: tCk'ljJ2} > O.
We first prove that (10.34) We argue by contradiction. Assume there exists some integer k such that tk = inf {t j ; 1 ~ j ~ m}
< 1.
Clearly Cj 'ljJ1 2: tkCj'ljJ2 ; 1 ~ j ~ m .
We note that N = 2:::7'=1 Cj and 'ljJ1
=
(0 - T)-l
",m
L..JJ =l
C j 'ljJ1 2: tk(O _ T)-l ",m Cj 'ljJ2 L..JJ =l
= tkN'ljJ2 = tk'ljJ2 · Thus N'ljJ1 2: N(tk'ljJ2) and Ck'ljJ1
= Ck N 'ljJ1
2: Ck N (tk'ljJ2)
2: (1 + J.L%)t kCkN('ljJ2)
232
Topics in Neutron Transport Theory
This contradicts the definition of tk. Thus, (10.34) holds and consequently
Hence
'l/J1 = (0 -
T)-l
2::=1 Ck'I/J1
~ (0 -
T)-l
2::=1 Ck'I/J2
=
'l/J2.
Changing the role of 'l/J1 and 'l/J2 yields the inequality 'l/J2 ~ 'l/J1 ' 0 To prove that CkN (1 ::; k ::; m) are I-concave, we need a technical result. Lemma 10.6 Let'I/J E B ,'I/J =I- O. Then, for all 0 < t < 1,
and is not identically zero. Moreover, if 'I/J
n x Vk,
> 0
a. e. on
nxV
then, on
Proof. Let Fk(Z) be defined by (10.10). We note that it vanishes at Z
= 0 and that
Let z E [0, I]k be fixed. We set h(t) := Fk(tZ) ; t E
[0, 1]
and note that
is nonincreasing in general and is decreasing if z > 0 (Zi > 0, Vi) . Using h(O) = 0 th' (t) - h(t) t2
233
Chapter 10. Nonlinear problems
th' (t) - (h(t) - h(O)) t2
th' (t) Hence (¥)' ~ 0 and, if general,
Z
>0
(Zi
~/h' (Bt)
(0
< B < 1).
> 0, Vi), (¥)' < 0 on )0, 1) . Thus, in (10.35)
and, if
Z
> 0, Fk;tZ) > Fk(Z) ;
t E
)0, 1[ .
This ends the proof by choosing Zi = 'lj;(x, v~) (1 ~ i ~ k). We complement the previous uniqueness result by
(10.36)
0
Theorem 10.7 We assume that (10.29) is satisfied and there exist Ak > 0 (2 ~ k ~ m) such that
then CkN (1 ~ k ~ m) are I-concave . Proof We begin with CIN. Let 'lj; E B, 'lj; =1= 0 and 0 < t < 1. We have to prove the existence of Ql, f31, III > 0 such that
(10.37) The existence of f31 is clear. We have m
N('lj;) = (0 - T)-1
L Ck'lj; k=1
so that
C 1 N('lj;) ~ C 1 (0 - T)- I C1 'lj;.
The arguments used in the proof of Theorem 10.5 show that C 1 (0-T)-lC1'lj; has a positive lower bound Ql . The first part in (10.37) amounts to
Topics in Neutron Transport Theory
234
Hence it suffices to prove the existence of d l
> 0 such that
cdJ"(t'l/J) - tClN('l/J) ~ dl and to choose III small enough. By linearity of Cl
(10.38) ,
m
ClN(t'l/J) - tClN('l/J) =
L
[Cl(O - T)-lCk(t'l/J) - tCl(O - T)-lCk('l/J)].
k=2 Moreover, Cl(O - T)-lCk(t'l/J) - tCl(O - T)-lCk('l/J) is equal to Cl(O - T)-l [Ck(t'l/J) - tCk('l/J)]
and Ck(t'l/J) - tCk('l/J) is equal to
/Vka(v)ck(X,V,V~, .. ., v~) [1-Il~=1(1-t'l/J(x,v~))] { - / Vk a(v)ck(X, v, v~, .. ., v~)t [1 Therefore
(10.39)
Il~=l (1 -
'l/J(x, v~)) 1 ~
o.
ClN(t'l/J) - tClN('l/J) ~ 0
in view of Lemma 10.6 and, for all k = 2, ... , m,
ClN(t'l/J) - tClN('l/J) ~ Cl(O - T)-l [Ck(t'l/J) - tCk('l/J)] .
(10.40)
Set
cp(x,v~, .. .,v~)= [1- rr(1-t'l/J(X'V~))l-t[1- rr(1-'l/J(X'v~))l · 3=1
3=1
We note, in view of Lemma 10.6, that cP side of (10.40) is equal to
=
/
n xvk
~
0, cp
i= O.
Thus, the right-hand
" , " , " , P (x" x ,v,vl' ... 'vk)cp(x ,vl , ... ,vk)dx dvl ·· · dvk
where P(x, ,x' , v, v~, ... , v~) is equal to 00
I
X- X ) e- u ( - . - Sa(v)cl(X, ~/
o
I
I
x-x x-x v, - - ) a ( - - ) s
s
(10.41)
235
Chapter 10. Nonlinear problems and
C1,
ck are extended by zero outside V. We note that
, , , P(x"x ,v,v1 "",vk) G(x-x'»O. Hence
where
\[!(x')
=
r cp(x',v~, ... ,v~)dv~ .. . dv~.
iVk Since \[! =1= 0, it follows that In G(x -
x')\[!(x')dx' > 0 (and is continuous by a convolution argument). Finally,jIO.38) follows in view of (10.40). We consider now the I-concavity of CkN for k ;::: 2. Let 'l/J E B, 'l/J =1= 0 and o < t < 1. We look for O!k , Ok , /-Lk > 0 such that CkN(t'l/J)
> (1 + /-Lk)CkN('l/J) (10.42)
The existence of Ok is clear. On the other hand, k
CkN('l/J) = Ck(O - T)-l
L Cj'l/J ;::: Ck(O - T)-lC 'l/J 1
j=l and Ck(O - T)-lC 1 'l/J is equal to
r a(v)ck(X, v, v~, ... , v~) [1- j=l IT (1- (0 - T)-lC 'l/J(x, V~))l, dv~ '" dv~ 1
iVk
so that
C,N(.p)
?y J.,
L
[1-
g
1
(1 - (0 - T)-'C,1>(x, v;)) dv; .. dv;
By the normalization assumption
IVk dv~ . .. dv~ =
1,
236
Topics in Neutron llansport Theory
where
p(x) =
L(o
- T ) - ~ C I $ ( vX1,) d v ' .
Arguing as previously, one sees that p ( x ) has a positive lower bound, whence the existence of a k follows. To prove the first part of (10.42), it suffices to prove the existence of dk > 0 such that:
We note, in view of (10.39),that
We set
k
p := (0 - T ) - l
>
z ~ j (( 0 $ -T ) )-'c~$.
j=1 Then
>
c k N ( t $ ) - t ~ k f i ( $ ) C k ( t p ) - tCk ('PI.
On the other hand, C k ( t p )- t C k ( p ) is equal to
and therefore
where @ ( x ,v ; , ...,v ; ) is equal to k
1-1
-(
j=1
Thus, using the fact that
X V ) )
(10.44)
237
Chapter 10. Nonlinear problems is nondecreasing and (10.44), we obtain
with
cp(x) =
Iv
(0 - T)-lC1'l/;dv .
As in the previous case, one sees that cp is bounded away from zero and therefore (10.43) is satisfied. <:;
10.8
The time dependent problem
We study now the existence of global solutions in B for the time dependent problem (10.3) which can be viewed as a Cauchy problem
d'l/;
dt
= T'I/;
+ N'I/; ; '1/;(0) =
We note that T generates a sernigroup {S (t); t r.p E LOO(n x V)
--+
(10.45)
'1/;0 E B. ~
o} in L 00 (0, x V)
e-u(v)tr.p(x + tv, v)xn(x + tv)
which is not strongly continuous. Instead of (10.45), we deal with its integral version 'Ij;(t)
= S(t)'Ij;o +
lot S(t - s)N'I/;(s)ds , 'I/;(t)
E
B.
(10.46)
We define the space X as
{J: [O,oo[
--+
LOO(n x V); w*- continuous and bounded}
and B := {J E X; f(t) E B, t E [0, oo[} .
Theorem 10.8 Let '1/;0 E B . Then (10.46) has a unique global solution in B. Proof We define the map
N: f E B
--+
S(t)'Ij;o +
lot S(t - s)N'I/;(s)ds .
Topics in Neutron llansport Theory
238 It is easy to see the continuity of
t
+
( N f( t ) g) 1
where g E L1(R x V) and (., .) is the duality pairing, that is %f(.) is w*continuous. On the other hand,
and Hence
and therefore W f ( t )E B for all t 2 0. Let cp, $ E B and let ? > 0 be fixed. Using the contraction property of { S ( t ) ;t 2 O),
and
where M is the Lipschitz constant of the polynomial operator N o n B since cp(s), $ ( s ) E B. Thus, we get existence and uniqueness on the time interval [o, Since the lifetime does not depend on the initial data in B, we can extend the solution by standard arguments. This ends the proof. 0
$1.
10.9
Time asymptotic behaviour
This section deals with the study of the limit as t -+ +oo of the time dependent solutions. To this end we give another approach of (10.46) based on monotonicity arguments. Theorem 10.9 Let cp E B and define the inductive sequence {+k)
(i) If cp is a subsolution, i.e. cp 5 %cp, then {$k) is nondecreasing and converges pointwise to the solution of (10.46) . (ii) If cp is a supersolution, i.e. cp 2 then {$k) is nonincmasing and converges pointwise to the solution of (10.46).
rep,
239
Chapter 10. Nonlinear problems
Proof We note that
We already know that N leaves B invariant. By using the fact that N is nondecreasing and that S(t) is non-negative the claim follows by standard arguments used previously. <> Before proceeding further we give a preliminary result. Lemma 10.7 The subspace Xex> of X consisting of those maps having a limit in Lex>(n x V) norm as t -+ +00 is invariant under N. More precisely, if cp E Xex> then Ncp E Xex> and lim Ncp(t) = (0 - T)-l [ lim cp(t)]. t-++ex> t-++ex>
Proof Let cp E Xex> and let CPex> = limt-++ex> cp(t). We have Ncp(t) = S(t)'l/Jo We note that S(t)'l/Jo
J:
-+
+ lot S(t - s)Ncp(s)ds .
0 in Lex> as t
S(t - s)Ncp(s)ds
=
-+
J:
+00 and
S(s)Ncp(t - s)ds
= J~ X[O,t] (s)S(s)Ncp(t - s)ds . In view of the continuity of N in Lex>(n x V), X[O,t] (s)S(s)Ncp(t
- s)
-+
S(s)N(cpex» ,
Moreover
loex>
IIS(s)11 ds < 00.
Hence, by the dominated convergence theorem,
and this ends the proof. <>
\:j
s > o.
240
Topics in Neutron Transport Theory
Theorem 10.10 If ru [(0 - T)-l K] < 1, or if ru [(0 - T)-l K] (10 .20) holds, then 'ljI(t) --+ 0 in LOO(O x V) as t --+
=
1 and
+00
where'ljl(.) is the solution of the Cauchy problem (10.46) . Proof. We define the inductive sequence {¢d, ¢k+l = N'ljIk, ¢o = 1. Then, according to Theorem 10.9, {¢d is nonincreasing and converges pointwise to 'ljI( .). Moreover, according to Lemma 10.7,
It follows, by induction, that
(10.47)
where {1pd is defined inductively by
One sees that {~d is nothing but the sequence used in Theorem 10.1 to prove the existence of a maximal solution to the stationary problem, and
where
~
is the maximal solution. Hence
and, in view of (10.47) ,
This ends the proof because ~ = O. <> The treatment of the supercritical case is more technical, and we need a preliminary result. Let 'ljI* be a non-negative eigenfunction of (0 - T)-l K corresponding to its spectral radius introduced in Section 10.6, normalized by 11'ljI* IILOO(O XV ) = 1 and 'Po = c'ljl*, with c::; co (see (10.25)) . According to (10.26), (10.48) We have
241
Chapter 10. Nonlinear problems Lemma 10.8 Let ra [(O-T)-lK]
> 1. We assume that the initial data
1/Jo of the Cauchy problem (10.46) is bounded away from zero. Let z = inf 1/Jo· 1 If c ~ ).* IIKII- z then CPo ~ Ncpo, i.e. CPo is a subsolution of N. Proof Let us prove that N CPo - CPo 2: 0, i.e.
+ lot S(s)Ncpods - CPo 2: O.
S(t)1/Jo
According to (10.48), it suffices to show S(t)z + lot S(s)Ncpods - (0 - T)-l Ncpo 2: 0 ,
i.e. S(t)z
-1
00
(10.49)
S(s)Ncpods 2: O.
Let (x, v) E n x V. For t > s(x, v), each term in (10.49) vanishes. If t < s(x, v), then (10.49) reduces to e-ta(v) z -
J~ e-sa(v) Ncpo(x + sv, v)xn(x + sv)ds
= J~ e-sa(v) On the other hand, N CPo
[u(v)z - Ncpo(x ~
+ sv, v}xn(x + sv)] ds.
K CPo and
u(v)z - cK1/J*(x + sv, v) 2: )'*z - c IIKII 2: 0
and the proof is complete.
<>
Theorem 10.11 Let ra [(0 - T)-l K] > 1. We assume that the initial data 1/Jo of the Cauchy problem (10.46) is bounded away from zero and that the nontrivial solution of the stationary problem V5 is unique, then
where 1/J(.) is the time dependent solution. Proof We define the inductive sequence {1/J k }, 1/J k -
-
+1
= N1/J-k'1:..o .1. = CPo .
{tk}
According to Lemma 10.8, CPo ~ Ncpo. By using Theorem 10.9, is non decreasing and converges pointwise to 1/J(.). According to Lemma 10.7,
t1(t)
--t
(O-T)-lNcpo in LOO(n x V).
Topics in Neutron Transport Theory
242 It follows, by induction, that
where the sequence {!ek}' defined inductively bY!ek+l
= (O-T)-lN!ek ':£0 =
'Po, is nothing but that used in Theorem 10.4 to prove the existence of a
nontrivial solution. Thus
where (~d is the sequence introduced in the proof of Theorem 10.10. Hence
111fJ(t) -
This ends the proof since both {
10.10
Comments
The material in this chapter was taken from K. Jarmouni-Idrissi and M. Mokhtar-Kharroubi [5]. However, Lemma 10.4 is not given there and the convergence results of the last section are obtained there for almost all (x, v) E n x V. The first mathematical treatment (i.e., existence, uniqueness and asymptotic behaviour) of the generating function is due to A. pazy and P. Rabinowitz [12] [13] [14] for a model case where all the parameters are constants. The extension of [12] to the general case under the assumption of non-correlation, i.e.
is due to M. Mokhtar-Kharroubi [9]. Several open problems are worth mentioning. First, the theory above is intimately connected to the existence of leading eigenelements of linearized transport operators which provide us with subsolutions of the nonlinear problems. Thus, in unbounded domains,
Chapter 10. Nonlinear problems
243
the main problems are open (see K. Jarmouni-Idrissi [6] for partial results) . The convexity of the spatial domain plays a basic role in many places, in particular to prove uniqueness results. An extension of the previous results to non convex domains would be useful. The different assumptions on the parameters Ck (.) could be, probably, relaxed to some extent. However, we mention that multiple nontrivial solutions may exist if the strict positivity assumptions are too much relaxed [9]. It would be useful to study more systematically the set of nontrivial solutions. In the supercritical case, the convergence of the time dependent solution to the nontrivial stationary solution is based on both the uniqueness of the latter and the assumption that the initial condition be bounded away from zero. The understanding of the time asymptotic behaviour, when such assumptions are weakened, is a very interesting open problem. The derivation of the equations governed by various moments (first, second ... ) of the generating function can be found in M. Otsuka and K. Saito [10] and G.!. Bell [1] (see also K. JarmouniIdrissi [6]). The derivation of the equations governing the probabilities Pj (t f , x, v, t) (j = 0, 1, .. .), and their mathematical analysis are given in D. Verwaerde [15] [16]. We refer to G.!. Bell [1] for the various ramifications of the stochastic neutron transport theory and for extensive references. More recent references may be found in J . Lewins [7]. Nonlinear problems involving monotone operators (in the lattice sense) have been extensively studied in the mathematical literature (see H. Amann [2], P.L. Lions [8] and references therein). We refer also to E. Zeidler [17] for a nice introduction to the subject and for a comprehensive bibliography. We mention also general functional analytic results in abstract cones in Dajun-Guo and V. Lakshmikantham [4].
References [1] !. Abu-Shumays et al (Ed). Transport Theory. SIAM-AMS Proc. Providence. Rhode Island, 1969. [2] H. Amann. Fixed-point problems and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976) 620-709. [3] G.!. Bell. Stochastic theory of neutron transport. Nucl. Sci. Eng. 21 (1965) 390-401. [4] Dajun-Guo and V. Lakshmikantham. Nonlinear Problems in Abstract Cones. Academic Press, 1988. [5] K. Jarmouni-Idrissi and M. Mokhtar-Kharroubi. A class of nonlinear problems arising in the stochastic theory of neutron transport.
Topics in Neutron Transport Theory
244
Nonlinear Analysis. (1997) . To appear.
[6J K. Jarmouni-Idrissi. Etude d'une classe de problemes non lineaires provenant de la modelisation probabiliste des chaines de fission neutroniques. These de l'universiU de Franche-ComU, Besan<;on, 1997.
[7J J . Lewins. Linear stochastic neutron transport theory. Proc. R. Soc. Lond. A 362 (1978) 537-558. [8J P. L. Lions. On the existence of positive solutions of semilinear equations. SIAM Rev. 24 (1982) 441-467. [9J M. Mokhtar-Kharroubi. On the stochastic nonlinear neutron transport equation. Proc. Roy. Soc. Edinburg. 121 A (1992) 253-272.
[lOJ M. Otsuka and K. Saito. Theory of statistical fluctuations in neutron distributions. J. Nucl. Sci. Technol. 2(8) (1965) 304-314.
[I1J L. Pal. Statistical theory of neutron chain reactions. Acta. Phys. Hung. 21 (1962) 390-. [12J A. Pazy and P. Rabinowitz. A nonlinear integral equation with applications to neutron transport theory. Arch. Rat. Mech. Anal. 32 (1969) 226-246.
[13J A. Pazy and P. Rabinowitz. Corrigendum. Arch. Rat. Mech. Anal. 35 (1969) 409-410.
[14J A. Pazy and P. Rabinowitz. On a branching process in neutron transport theory. Arch. Rat. Mech. Anal. 51 (1973) 153-164. [15J D. Verwaerde. Une approche non deterministe de Ie Neutronique : ModeIisation. Note CEA N 2731, 1993.
[16J D. Verwaerde. Une approche non deterministe de Ie Neutronique. Existence d'une solution aux equations de probabilite de presence. Note CEA N 2791,1995.
[17J E. Zeidler. Nonlinear Functional Analysis and its Applications, Vol 1. Fixed point theorems. Springer Verlag, 1986.
Chapter 11
Velocity averages and inverse problems 11.1
Introduction
The role of velocity averages in the analysis of the compactness in Transport theory and its consequences for spectral theory was fully emphasized in Chapters 3 and 4. We are concerned, in this chapter, with the use of velocity averages in the context of inverse problems. More precisely we will show how the knowledge of moments (velocity averages) of the solution of transport equations, with respect to suitable measures, can be used to determine explicitly the spatial part of internal sources. We consider stationary equations of the form
J
v.~: +a(v)~(x,v)
k(x, v, v' )~(x, v')dm(v')
+ S(x, v)
v (11.1) where (x, v) E n x V; n c Rn is a smooth open convex subset and V is the set of velocities endowed with a positive measure dm. As usual,
r_=
an x V ;v .n(x) < O} where n(x) is the outward normal at x E an. We first consider collisionless {(x,v) E
transport equations of the form
v.
a~
ax
+a(v)~(x,v)
245
S(x,v)
Topics in Neutron Transport Theory
246
(11.2) The treatment of inverse problems for (11 .1) can be deduced easily from that of collisionless transport equations. Fairly general results hold in one dimension while, in higher dimensions, we will need, for technical reasons, some additional assumptions such as homogeneous internal sources and zero boundary sources. We close this introduction by explaining the main idea behind the mathematical analysis of this chapter. In dimensions n ~ 2, if the source does not depend on velocities, 'P- = 0 and if dp, is a given (signed) measure on V, then the moment
is related to the source by a convolution integral equation 'Pl(X)
=
In
N(dp, ; x - y)S(y)dy
where the kernel depends on the measure dp,. The key idea is that , if dp, is chosen in an appropriate class C of measures then
6. [N(dp, ; .)]
= N(d{3; .) + cnDo
where 6. denotes the Laplacian in the sense of distributions, Do is the Dirac mass at the origin, Cn is a constant and d{3 is another measure given explicitly in terms of dp,. It follows that
where 'P2( X)
=
Iv
'P( x , v)d{3(v) .
Thus, the source is recovered explicitly in terms of two moments of the solution provided that Cn =f. O. We note that the solution itself is then recovered explicitly in terms of two of its moments. We will see, for n =f. 2, that there exists a general subclass of C for which Cn =f. 0, while C2 = 0 for all dp, E C. The extension of the inverse results to (11.1) is straightforward for scattering kernels of the form k(x, v, v' ) = k(x , v'). As a by-product, for known sources, it is possible to determine the spatial part of scattering kernels of the form k(x , v , v' ) = k 1 (x)k 2 (v').
247
Chapter 11. Velocity averages and inverse problems
11.2
One dimensional inverse problems
Let -00 :'S a < b :'S +00 . We are concerned with the following boundary value problem
J 1
k(x, (.k, (.k')-t/J(x , (.k')dm((.k' ) + Sex, (.k)
-1
We first consider the case of purely absorbing media
which is solved (formally) explicitly, according as (.k
> 0 or (.k < 0, by
J x
= e-lW(x-a)g+((.k) + I~I
't/J(x ,(.k)
e-lWIX-X'IS(x',(.k)dx'
a b
=
) 't/J ( X,(.k
e -~(b-x) 11'1 g- ((.k)
1 _ I-~Ix-x'i +~ e 11'1 S(' X ,(.k)dx ' (11.5) x
where the term involving g+((.k) (resp. g-((.k)) is to be dropped if a = -00 (resp. b = +00) . Let dO'. be a bounded measure on J-1, +1[ (not necessarily positive) satisfying the following conditions dO'. is invariant by symmetry with respect to zero {
J l
o
~ < 00
(11.6)
1-'2
where dial is the absolute value of dO'. . The role of those conditions will appear clearly in the sequel. We recall (see [6J [7]) some trace results. Let 1 :'S p < 00 and let
Topics in Neutron Transport Theory
248
x = V(]a,b[ x ]-I,+I[;dx ® dla!) and
W=
{7/JEX;Ik~~ EX} .
Each element 7/J of W has traces on the spatial boundary such that
1:1Ilkll7/J(a,lkW dial (Ik) < +00 (if a is finite) 1 1: Ilkll7/J(b, IkW dial (Ik) < +00 (if b is finite) . Thus, to solve (11.4) in W, we assume g- E V(]-I, 0[; Ilkl d la!) ; g+ E V(]O, 1[; Ilkl d la!)
!
a E L=(]-I, 1[; d la!) ; ess inf a( .) =
,\*
>0
(11.7)
S E V(]a, b[ x ]-1 , +1[; dx ® d la!) .
The following evenness assumptions are crucial for the sequel
(11.8) and
S(x, Ik) = S(x, -Ik) dial a.e.
(11.9)
We state the elementary result Proposition 11.1 We assume that condition (11.7) is satisfied. (11.4) has a unique solution in W given by (11 .5) .
Then
The main result is the following Theorem 11.1 Let (11.6)-(11.9) be satisfied and let 7/J denote the solution of (11 .4). We assume that
is finite. Then
(i) 7/J E V(]a, b[ x ]-1, +1[ ;dx ® ~)
249
Chapter 11. Velocity averages and inverse problems
(ii) IPl(') =
+1 /
-1
'f/;(.,/J)da(/J) E W 2,P(]a , bD and
Before giving the proof, let us derive some direct consequences.
Corollary 11.1 Let the conditions of Theorem 11.1 be satisfied. If S(x, /J) = +1
Sl(X)S2(/J) and if.[
-1
a(/J)S2(/J)d:\f)
I- 0,
then
where
Remark 11.1 Let d{J =
a( 1L)2
~da .
knowledge of the moments
/
explicitly Sl(X) provided that
Then Corollary 11.1 expresses that the
+1 'f/;(x, /J)da(/J) -1
and
/+1 'f/;(x, /J)d{J(/J) yields -1
+1 /
a(/J)S2(/J)do:\j') is known. Note that the -1
!Jo
determination of Sl does not depend explicitly on the incoming distribution (g-,g+) . The choice of evenly spaced Dirac measures yields
Corollary 11.2 Let /Jo E ]0, 1[ and let da = D!Joo
+ D-!Joo'
Then
2
S(x, /JO) = [IP2(X) where IPl(X) = 'f/;(x,/Jo)
+ 'f/;(x, -/Jo)
IP~ (x)] 2~~o)
and IP2(X) = ~IPl(X). !Joo
(11.10)
Topics in Neutron Transport Theory
250
Remark 11.2 As a consequence of Corollary 11.2, one sees that, if 0'(.) and S( x , .) are even then tp(x, v) = 1jJ(x , v) + 1jJ(x, -v) satisfies the second order equation
Thus we find again the even formulation of the transport equation [30] . Proof of Theorem 11.1 : The part (i) is a simple consequence of Proposition 11.1 where we use the measure ~ instead of dial. Consider now part (ii) . The solution 1jJ is given by (11.5) so that, by the evenness assumption, tp1(X) is equal to
J J 1
0
1jJ(x, Ji-)da(Ji-)
+
o
J
1jJ(x, Ji-)da(Ji-)
-1
1
o
+
e-lW(b-x)g_(Ji-)da(Ji-)
-1
J Je-l'Wlx-x'IS(x"Ji-)da~Ji-) b
+
J 0
e-lW (x-a)g+ (Ji-)da(Ji-) 1
dx'
a
.
(11.11)
0
The last term is given by
It follows that tp~ (x) is equal to
-I: (1~e)e-l'W(x-a)g+(Ji-)da(Ji-) +
J:
-I:
S(X,Ji-)daje) -
S(x , Ji-)daje)
+
+ J~l
'~l)e-lW(b-x)g_(Ji-)da(Ji-)
J: J: '~l)e-lW(x-x')S(X',Ji-)d,jr) J: J: (11~l)e-lW(x'-x)S(X" Ji-)d,jr) · dx'
dx'
251
Chapter 11. Velocity averages and inverse problems Differentiating again, one sees that
-J: cr£~)S(x,p,)da(p,) I: J: -J: J: J: cr£¥) S(x, p,)da(p,)
=
1crl,,\2 Jo
~(x
~e- II'I
-
a)
+
dx'
cr<;'l e-lW(x-x' )S(x' , p,)di£r)
+
dx'
cr(:/ e-lW(x'-x) S(x'
(
9+ p,)da(p,)
+
JO -1
/l.
crl,,\2
~(b x)
~e- II'I
-
,p,)di£r)
9_(p,)da(p,)
/l.
We observe that
(see (11.11) where 1j; is integrated with respect to the measure da) . Hence
and the proof is complete. <> The extension to scattering media is straightforward. Indeed, consider the problem 1
J
k(x, p,')1j;(x, p,')dm(p,') + S(x)
-1
(11.12)
252
Topics in Neutron l3ansport Theory
where the scattering kernel and the source are assumed to be independent of the yvariable. We set
According to Corollary 11.1, the knowledge of two m~mentsof the solution $ of (11.12) determines explicitly the right-hand side S ( x ) and consequently the solution II, itself through (11.5), where S is to be replaced by g. This gives the term
since k ( z , {) is known. Finally we deduce the source S ( x ) from (11.13) . We leave to the reader the formal statement of this result. Assume now that k(x, = k1( x ) k 2 ( ~and ' ) the source S ( x ) is known while k l ( x ) is unknoum. Then, by the preceding arguments, the knowledge of two moments of the solution $ of (11.12) determines S and the solution $ and consequently the moment
so that we can determine the cross-section k l ( x ) by means of the relation
11.3 Multidimensional inverse problems This section deals with the dimensions n 2 3. We restrict ourselves to purely absorbing media; the extension to scattering media is straightforward as in the preceding section. We consider the transport equation
where ( x ,v ) E R x V, R is a smooth convex open subset of Rn and
Let
Chapter 11. Velocity averages and inverse problems
253
For technical reasons, we assume that 'IjJ- =0
Ivl
u(v) = u(p) ; p =
1
(11.15)
u(.) E W 1,OO([0, 1]) ,
i.e. u( .) is radial and Lipschitzian and there is no incoming source. Assuming that infu(.)=A*>O, the solution of (11 .14) is given by r(x,v)
'IjJ(x, v)
= Jo
e-sa(v)S(x-sv)ds
where
s (x, v) = inf {s >
(11.16)
°;
x - sv tJ. n} .
Let df.L(v) = da(p) ®dT(W) be a (signed) measure on V where dT(W) is the surface Lebesgue measure on the unit sphere sn-l and da(p) is a (signed) measure on [0,1) satisfying the conditions
1 1
da(p) =
o
° 11 dial ,
PoP
(p)
2
<00
(11.17)
where dial (p) is the absolute value of da(p). Those conditions (particularly the first one) are going to playa crucial role in the sequel. We note also that positive measures are excluded contrary to the one dimensional analysis of the previous section. We introduce the function of bounded variation
k(p) =
r da(s) Jo
s
and the bounded measure
where dp is the Lebesgue measure on [0,1] . The main result in this section is the following Theorem 11.2 We assume that (11.17) is satisfied. Let'IjJ be the solution of (11.14) and let
=
11
da(p)
fsn-, 'IjJ(X,pw)dT(W)
Topics in Neutron llansport Theory
254 and M ( X )=
1' Ln-L
$(x, ~ ) d r ( ~ ) .
Then, denoting by A the Laplacian in the sense of distributions, (2) A p 1 E P ( 0 ) (22) PI = knS(x)+ ~ ( 2 ) where kn = (n- 2)
Isn-' I
/
1
Yda(s). 0
Remark 11.3 As in the previous section, the knowledge of the moments pl and p2 of the solution determines explicitly the source term provided that J: y d a ( s ) is not equal to zero. Proof: Let $ be the solution o f (11.14) which is given, in view of (11.16), by
where v = p.d and w E Sn-l. Hence
Making the change of variables x - sw = x', using dx' = sn-ldsdr(w) and the convexity of 0, we obtain
Let k be the primitive of
vanishing at the origin, i.e.
Chapter 11. Velocity averages and inverse problems and let
q(p) =
255
e -~u p un - 1
where we have dropped the u-dependence for the simplicity of notations. Then, integration by parts (for Stieljes measures) gives
.I:
.I:
=-
q(p) daJe)
k(p)q' (p)dp + [k(l)q(l) - k(O)q(O)] -~u
1
/(p)(_~)'eun~2 dp+k(l) e::~;u
=/
because
k(l) = (I da(p) = 0
Jo
p
in view of (11.17). Let
z(p) = k(p)(P))'. P
Then (11.19) becomes
1
/
[
/
z(p) (
e-
p
Ix-x
0
fl
~Ix-x'i
1
,
-1
n-2
I
)
1
"
dp Sex )dx
,
+/Z(P)dP / o fl
S(x)n
Ix - x'l
2 dx ' .
(11.20)
-
Before investigating further (11 .20), we consider the radial function
u E Rn
-t
F(p, u) =
whose Laplacian in R n 6 u F(p, u)
=
-
e -~I'U.I p
lui
n
-
1
e -~r p
2
{O} is given by
"
n-1 r
f (r) + --f'(r)
rn -
2
-
1
= fer)
;r
= lui
Topics in Neutron llansport Theory
We write now (11.20) in the abstract form
Taking the Laplacian of (11.22) (in the sense of distributions) yields
since
1 (2 - n) ISn-ll
1
~
~
l
~
-
~
>
is the fundamental solution of the Laplacian in Rn (n 3). In view of (11.21). the first term on the right-hand side of (11.23) is given by
We focus our attention on the second term of (11.24) . It is equal to
257
Chapter 11. Velocity averages and inverse problems
We observe that
=
I
d
(2 (p)k(p) ) e - tiellx-x'i P
<7
_[ 0
=
1 /
o
Ix-x 'In
p2
d
(2 <7
(p)k(p) p2
) - tiellx-x'i e
[e tiellx-x'i 2
_
P
P
Ix-x 'In
(X
•
i
<7
Ix-x 'In
i
-If
,
i
(p~k(p) p
1
p=1 p=o
X')
because
k(l) = (I da(p) = O.
Jo
p
Hence (11.24) amounts to 1
tiellx-x'i
_
z(p)(n - 3)(J(P)dP / e P ,n-l S(x')dx' / pIX-X I o n
+/1 pd ((J2(P)k(P)) / P2
o 1
/
_
d(3(p)/e
o
n
e-
n
~Ix-x'i1 S(x')dx'
I 'I PX-X
n
-
tiel lx-x' I P
pIX-X
,n_ 1 S(x')dx'
I
where
d(3(p)
= z(p)(n - 3)(J(p)dp + pd (<72(p~k(P») = (n - 3) [(J(p) (p(J' (p) -
(J(p))~] dp + pd (<72(p~k(P»)
.
Finally (11.23) becomes 1
/
/
tiel lx-x' I
_
d(3(p)
e
,n-l
P
plX-x
I
S(x')dx'
o
n
+(2 - n)
Isn- 1(/ z(p)dp)S(x).
1 1
o
(11.25)
258
Topics in Neutron llansport Theory
In view of k(1) =
So' ikfd = 0, an integration by parts shows that
By noting that the first term in the right-hand side of (11.25) is nothing but rl
r
the proof is complete. 0 Because of the peculiarity of the fundamental solution of the Laplacian , technicalities of the proof in two dimensions are slightly different. in R ~the We treat this case separately.
11.4
The dimension two
Our assumptions and notations are the same as in the preceding section. Let dB be the bounded measure
We note that it agrees with (11.18)when we set n = 2. The two dimensional version of Theorem 11.2 is the following Theorem 11.3 Let n = 2 and let the conditions of Theorem 11.2 be satisfied. Let $ be the solution of (11.14) and let
Then Apl E Lp(R) and
where dp(p) is given by (11.26). Proot According to the calculations in the proof of Theorem 11.2,
Chapter 11. Velocity averages and inverse problems
=
259
J1(~)'k(p)e-~IUldP' o p
The Laplacian of the radial function ('Y is a constant)
u E R2
-->
e-,.Iul
is given by
C'Y 2
J:J )e-,.Iu l .
-
It follows that the Laplacian (in the sense of distributions) of 1
2
uER
-->
1 0
e -~Iul p
pJuJ
da(p)
is given by
!\a(p))'k(p) [t(P))2 _ a(p)] io p p pJuJ
e-~Iuldp.
(11.27)
We decompose (11.27) as
(11.28) The first term of (11.28) is equal to
260
Topics in Neutron Transport Theory
Hence
~u ( 10
1
-~Iul ) e p da(p) =
~ul
10
1
-~Iul e p d{3(p)
~ul
where d{3(p) is given by (11.26) . Finally
I [1
=. n
=
1 0 e
-~Ix-x'i
plx-x'i
11 1 o
d{3(p)
1
d{3(p) S(x')dx'
'ljJ(x, pw)d7(W)
Si
and this ends the proof. <)
11.5
Characterization of the range of integrated fluxes
We restrict ourselves to n-dimensional problems (n ~ 3) . We introduce the space Bp(n) = {
10
lSi
t
10
d{3(p) f 'ljJ(X,pw)d7(W)
lSi
is sufficient to recover explicitly the source term Sex). We recall that
The present section is devoted to the characterization of the range of the operator S E IJ'(n) -+ (
Isn- 1 1 fl a(~) da(s) i= o.
10
s
Chapter 11. Velocity averages and inverse problems We note that
is a bounded operator from P ( R ) into P ( R x V; d x 8 d la1 (p) 8 dr(w)) and, also, from P ( R ) into P ( R x V; d x 8 d 1PI (p) 8 dr(w)). We denote it by (the same symbol) Q. It is merely the solution operator to (11.14) with homogeneous boundary condition. Let
and
According to Theorem 11.2, EIQS E Bp(R) so that the following operator is well-defined
We are ready to state Theorem 11.4 Let (971, 972) E Bp(R)x P ( R ) . Then there exists S E P ( Q ) such that
where II, is the solution of (11.14) with homogeneous boundary condition and source term S, if and only i f (cpl, 972) is an eigenvector of R associated to the eigenvabe 1. I n such a case S is given by A'pii'p2. Proof: Let (971, 972) E Bp(R) x P ( R ) . Then, according to Theorem 11.2, if such an S exists, it is given by Thus, the problem amounts to
with
262
Topics in Neutron llansport Theory
and this ends the proof.
0
Remark 11.4 A similar characterization holds in the presence of a scattering operator with kernel k(x, v, v') = x(x, v'). The details are left to the reader.
11.6
Comments
The results in this chapter were partly announced in M. Mokhtar-Kharroubi [23] [24]. The results were extended to transport operators on the torus by M. Mokhtar-Kharroubi and A. Zeghal[25]where the use of Fourier analysis shows the optimality of the relevant assumptions. These ideas were also a p plied by A. Zeghal [32] to time dependent problems in order to determines sources or initial datum. The formalism introduced in this chapter deserves to be systematized in order to deal with multidimensional problems with nonisotropic internal sources, boundary sources and nonwnvex domains. The impossibility to determine sources in dimension two by means of this formalism probably hides a mathematical peculiarity of this dimension which deserves further analysis. A similar phenomenon arises also in the determination of cross-sections from albedo operators [ll]. We note that Acpl E P(a)implies, according to Calderon-Zygrnund estimates (see, for when 1 < p < oo. instance, [16] p. 1023), that the moment cpl E W2~p(R) This very special regularity result is optimal on the torus [25]. The fact that we can recover the source by taking the Laplacian of a moment (for n 2 3) indicates that we cannot differentiate further (if the source is not differentiable !). On the other hand, in two dimensions, the Laplacian of a moment is equal to another moment (see Theorem 11.3). Does this mean that the smoothing effect of velocity averages is better in two dimensions ? We point out that the techniques given here are well suited to the determination of the spatial part of sources or cross-sections but are not usable for the determination of the velocity parts. The possible connections of the techniques introduced here with those of Radon Tkansform (see F. Natterer [26],[27]), also deserve to be investigated. Of course, many other inverse transport problems (involving different techniques) were investigated by different authors and the literature on this topic is quite important. We quote, without pretence of completeness, P.P. Abbati and Marescotti [I],K.M. Case [5], E.W. Larsen [20][21][22], N.J.Mc Cormick [12][13], N.J.Mc Cormick and R. Sanchez [14][15], K. Dressler [18][I91, Y. Bin and Y. Mingzhu [4], F.
Chapter 11. Velocity averages and inverse problems
263
Natterer [26], M. Choulli [8], M. Choulli and A. Zeghal [9], V.I. Agoshkov [2], G. Dong geng [17], L.B. Wertheim [31], S.K. Patch [28], V.G. Romanov [29]. An account of the literature in the late Soviet Union is given in Y.E. Anikonov [3] Chap 2. Inverse problems, in the context of scattering theory, are considered in M. Choulli and P. Stefanov [lo] (see Chapter 14).
References [I] P.P. Abbati and Marescotti. Un problema di trasporto dei neutroni entro un mezzo sferico. Estr. Dagli. Atti. Accad. Sci. Torino. 9 7 (1962-63) 1-16. [2] V.I. Agoshkov. An inverse problem of transport theory and properties of the reflection operator. Plenum. Publ. Corp. 27(6) (1991) 709712. [3] Y.E. Anikonov. Multidimemional Inverse and Ill-Posed Problems for Differential Equations. Inverse and Ill-posed Problems Series, Utrecht, 1995. [4] Y. Bin and Y. Mingzhu. A class of inverse problems in a Banach space and applications to Transport Theory. Preprint. [5] K.M. Case. Inverse problem in transport theory. Phys. Fluids. 16 N0lO (1973), 1607-1611. [6] M. Cessenat. Theorhmes de trace LP pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. 299 Serie I. (1984) 831-834. [7] M. Cessenat. Theorhmes de trace pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. 300 Serie I. (1985) 89-92. [8] M. Choulli. Determination of the spatialy-dependent scattering function for overspecified boundary conditions. h n s p . Theory Stat. Phys. 22(1) (1993) 97-107. [9] M. Choulli and A. Zeghal. Laplace transform approach for an inverse problem. Tramp. Theory Stat. Phys. 24(9) (1995) 1353-1367.
[lo] M. Choulli and P. Stefanov. Inverse scattering and inverse boundary value problems for the linear Boltzmann equation. Comm. Part. Dzfi Eq. 21(5&6) (1996) 763-785. [ll]M. Choulli and P. Stefanov. An inverse boundary value problem for the stationary transport equation Preprint (1996).
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[12] N. J. McCormick. Recent developments in inverse scattering transport methods. Dansp. Theory Stat. Phys. 13 (1984) 15-28. [13] N.J. McCormick. Methods for solving inverse problems for radiation transport. An update. Dansp. ~ h k o r yStat. Phys. 15 (1986) 759772. [14] N.J. McCormick and R. Sanchez. Inverse problem transport calculations for anisotropic scattering coefficients. J. Math. Phys. 22(1) (1981) 199-208. [15] N.J. McCormick and R. Sanchez. General solutions to inverse transport problems. J. Math. Phys. 22(4) (1981) 199-208. [16] R. Dautray and J.L. Lions (Ed). Analyse Mathematique et Calcul Numerique pour les Sciences et les Techniques. Tome 2, Masson, Paris, 1985. [17] G. Donggeng. A class of inverse problems in transport theory. Dansp. Theory Stat. Phys. 15(4) (1986) 476-502. [18] K. Dressler. Inverse problems, in der linearen Transport-theorie- ein neuer Zugang, Dissertation, Kaiserslautern, 1988. [19] K. Dressler. Inverse problems in linear transport theory. Eur. J. Mech. B/ Fluids. 8 (1989) 351-372. [20] E.W. Larsen. Solution of the inverse problem in multigroup transport theory. J. Math. Phys. 22(1) (1981) 158-160. [21] E.W. Larsen. Solution of multidimensional inverse transport problems. J. Math. Phys. 25(1) (1984) 131-135. [22] E.W. Larsen. Solution of three dimensional inverse transport problems. Dansp. Theory Stat. Phys. 17 (1988) 147-167. [23] M. Mokhtar-Kharroubi. Talk given in 13th International Conference on Transport Theory, Riccione (Italy), May 1993. [24] M. Mokhtar-Kharroubi. Inverse problems in transport theory. C.R. Acad. Sci. 318 Serie I. (1994) 43-46. [25] M. Mokhtar-Kharroubi and A. Zeghal. Inverse problems for periodic transport equations. Submitted. [26] F. Natterer. An inverse problem for a transport equation and integral geometry. Contemp. Math. 113 (1990) 221-231.
Chapter 11. Velocity averages and inverse problems
265
[27] F. Natterer. The Mathematics of Computerized Tomography. WileyThbner, 1986. [28] S.K. Patch. Consistency conditions upon boundary values for the Boltzmann equation. Talk given in Conference on Inverse Problems of Wave Propagation and Diffraction, INRlA, Aix-Les-Bains, France, September 1996. [29] V.G. Romanov. Stability estimates in problems of recovering the attenuation coefficient and the scattering indicatrix for the transport equation. J. Inv. nl-Posed Problems. 4(4) (1996) 297-305. [30] V.S. Vladimirov.Mathematical Problems in the One-velocity Theory of Particle Transport. Atomic Energy of Canada. Ltd Chalk River.Ont. Report AECL-1661(1963) . [31] L.B. Wertheim. An inverse problem for the system of kinetic equations resulting from the Vlasov equation for an equilibrium plasma. Sov. Phys. Dokl. 37(5) (1992) 224-225. [32] A. Zeghal. Problemes inverses et regularite en theorie de transport. These de Cuniversiie de Franche-Comie, 1995.
Chapter 12
Limiting absorption principles and wave operators in L (jL) spaces with applications to transport theory 1
12.1
Introduction
Roughly speaking, scattering theory deals with the analysis of time asymptotic (t ----> ±oo) equivalence of two dynamics governed, say, by two groups of operators {Uo(t) ;t E R} and {U(t); t E R} acting on (possibly different) Banach spaces. Typically, one dynamics is a suitable perturbation of the other considered as a simpler one. Scattering theory has a long tradition in the context of Schrodinger equations (see M. Reed and B. Simon [31] and T. Kato [19], [18] Chapter X) and hyperbolic operators such as wave equations or, more generally, symmetric systems (see P. Lax and R. Phillips [21] [22] and V. Petkov [28]) where the dynamics take place naturally in Hilbert spaces. The usual perturbations appear in the form of a potential, an obstacle (in which case the dynamics take place in different spaces) etc .. Another physical motivation arised much later, in 1975, when J. Hejtmanek [15] introduced a new class of scattering problems on Ll spaces, in the context of transport theory. This is the subject of this chapter. Referring to
267
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Topics in Neutron Transport Theory
{Uo(t); t E R} and {U(t); t E R} respectively, as the free dynamics and the interacting dynamics, it seems natural to think that small perturbations do not affect significantly the behaviour of the physical system for large times (a concept of smallness depending on the context). Thus, we may expect that the behaviour of {U(t); t E R} should be similar, in some sense, to that of {Uo(t); t E R} when t -+ ±oo. More precisely, when the dynamics take place in the same space, we expect that the perturbed solution U(t)f behaves like a free solution Uo(t)f- as t -+ -00, and behaves also like another free solution Uo(t)f+ as t -+ +00. The scattering operator is then defined as
8: f-
-+
1+.
To define rigorously this scattering operator it is expedient to define the socalled wave operators. Assuming, for instance, that we deal with bounded groups of linear operators with generators To and T in some Banach space X, we observe that lim IIU(t)f - Uo(t)f-II
t-.-CX)
=0
is equivalent to
f = lim U( -t)Uo(t)ft-t-OO
while lim IIU(t)f - Uo (t)1+11 = 0 t-++= is equivalent to
1+ = t-++= lim Uo( -t)U(t)f. Thus, it is natural to consider the wave operators
W_(T, To)
= s t-+-= lim U( -t)Uo(t)
; W+(To, T)
= s t-++= lim Uo( -t)U(t).
When such strong limits exist, the scattering operator may be defined as
Presented in such abstract way, scattering theory appears as a branch of perturbation theory of linear operators. It is intuitively clear that the existence of wave operators is tied to the fact that the two groups are, in some sense, close to each other or, which amounts to the same, the perturbation is, in some sense, small. One of the main problems in the mathematical scattering theory is to analyze the conditions (the smallness of perturbations) under which the wave operators exist. On the other hand, in some physical problems, the scattering operator
8:f--+1+
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Chapter 12. Limiting absorption principles
is physically observable, and it is taken for granted that a great deal of informations on the perturbations (the potential, the obstacle... ) are contained in this operator. Inverse scattering aims at extracting such informations (see V. Petkov [28]). This topic is dealt with in Chapter 14, in the context of transport theory. In the present chapter, devoted to direct scattering theory, we give an abstract scattering theory for positive groups acting on P(J1-) spaces in terms of limiting absorption principles and show, next, how this formalism fits into neutron transport groups in the whole space. The general framework is the following: Let X = L 1(J1-), where J1- is a positive measure (on some measure space) and let {Uo(t); t E R} be a bounded and positive group (with generator To) acting on L1(J1-). We consider a bounded positive operator
BE L+(L 1(J1-); L 1(J1-))
and denote by {U(t); t E R} the group with generator
T=To+B. We will deal with the following problems: (i) Under which conditions are the semigroups {U(t); t 2: O} or {U(t); t :S O} bounded? (ii) Do the strong limits slimt-dooU(t)Uo( -t) exist? (iii) Does the strong limit slimt---> +00 Uo (-t)U(t) exist? It will be shown that the answer to these questions is intimately tied to the existence of the strong limits
B(O± - To) := s lim B()" - TO)-1 A--->O± and to the size of their spectral radii. Moreover, it will turn out that the above problems are equivalent. We point out, in view of the boundedness of {Uo(t); t E R}, that
aCTo) C iR. On the other hand, a(To)nR (see [27] p. 295) . Hence
=1=
0 in view of the positiveness of {Uo(t); t
E
R}
o E aCTo) . Thus s limA--->o± ().. - TO)-1 do not exist in L1(J1-). One may interpret the existence of s limA--->o± B().. - TO)-1 as a limiting absorption principle where slimA--->o±().. - TO)-1 would exist as operators from L 1(J1-) into a larger space. Positivity as well as the choice of P space are crucial assumptions because of the additivity of the L1 norm on the positive cone. We already saw the importance of such assumptions in Chapters 8 and 9. We will
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Topics in Neutron Ti-ansport Theory
apply the abstract results to transport operators in the whole space where {Uo(t);t E R) represents the streaming group and B the collision operator, and show how the relevant abstract assumptions are expressed naturally in terms of properties of the different cross-sections.
12.2
Preliminary results
We begin with the following observation. Let X = L1 ( p ) .
Lemma 12.1 Let { W ( t ) ;t > 0 ) be a bounded positive q-semigroup on X with generator G and let C E L+(L1(p);L 1 ( p ) ) . Then (2)
lo rt
lim C t-+m
W ( s ) x d s exists for all x E X
(12.1)
i f and only i f Limx,o+ C(X - G ) - l x exists for all x E X . I n that case, we have
W ( s ) x d s = C(O+ - G ) - l x ; x E X
(12.2)
where C(O+ - G)-l denotes the strong limit s lirn~,~, C(X - G)-l (zi) Similarly, i f { W ( t )t; E R ) is a bounded positive q-group, then lim
t-+-m
c J!
W ( s ) x d s exists for all x E X
(12.3)
i f and only i f limx,o- C ( A - G ) - l x exists for all x E X . I n that case, we have
W ( s ) x d s = -C(O- - G)-'x ; x E X .
(12.4)
Proof: Since the positive cone L i ( p ) is generating (i.e. L1 ( p ) = L i ( p )L i ( p ) ) ,we may restrict ourselves to x E L i ( p ) . Let the strong limit (12.1) exist and denote it by C J : ~W ( S ) Z ~Then, S . for X > 0 ,
Since C(X - G)-l is nondecreasing in X > 0 it follows, by the monotone convergence theorem, that limx,o- C(X - G ) - l x exists and
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Chapter 12. Limiting absorption principles
Conversely, assume that C(O+ - G)-l exists. From the obvious inequality
we get
Letting>.
-t
0+, we obtain
By the monotone convergence theorem, we obtain the converse result and the reverse inequality
(+oo
C io
W(s)xds ~ C(O+ - G)-Ix .
This ends the first part of the Lemma. To deal with the second part, we where W(t) = introduce the positive bounded co-semigroup {W(t); t ~
O}
-
+00-
W( -t), with generator G = -G. According to the first part, C Io W(s)ds exists if and only if C(O+ - 0)-1 exists and these strong limits are equal. This amounts to
C
1°00 W(s)xds = -C(O_ - G)-Ix;
xEX
and the proof is complete. 0 The boundedness of the perturbed (semi)group plays a key role for the existence of wave operators. This point is clarified in the following Theorem 12.1 (i) We assume that the strong limit
B(O+ - TO)-l
:= s lim >'-.0+
B(>. - TO)-l
exists and that ru [B(O+ - TO)-l] < 1. Then T = To bounded positive co-semigroup {U(t); t ~ O} . (ii) If, in addition,
B(O_ - TO)-l exists and ifru [B(O_ - TO)-l] co-group {U(t); t E R}.
:= s lim >.-.0_
+B
generates a
B(>' - TO)-l
< 1, then T = To + B generates a bounded
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Topics in Neutron a m s p o r t Theory
To prove this theorem we need a technical preliminary result. We define the space H L of strongly continuous mappings
z : t E [O, +oo[ + Z ( t ) E L ( L ~ ( P L) ;~ ( P ) ) such that suptlo IIZ(t)ll < +oo, endowed with the norm
llzll := St Ul 0P IIZ(t)ll .
(12.6)
It is easy to see that H& is a Banach space for the norm (12.6). We introduce the operator
where the integral (12.7) is understood as a strong one. We prove first
Lemma 12.2 The operator L is bounded in H& and
ru(L) 5
[B(o+- ~ o ) - l ].
Proof: Let x E X+ and Z E H&. From
we derive the estimates
In view of the additivity of the L1 norm on X+,
I l B ~ o ( ~ ds ) ~= ll
1 Jot
~uo(s)xdSII.
Hence, using Lemma 12.1,
= llZll IIB(o+ - TO)-~XII I llZll IIB(O+ - To>-lII llxll.
If x E X is arbitrary, by decomposing it into positive and negative parts, IILz(t>xllI llzll IIB(O+ - ~ 0 ) - ~ ~ ~ [ 1 + 1 ~11+ 2-11 111 = IlZll IIB(o+ - To)-lIJ llxll .
Chapter 12. Limiting absorption principles This shows that LZ E H& and IILIILc
HL) I IIB(O+ - T ~ ) - l ( l .
For each integer n, a computation shows that LnZ(t) is equal to
Hence, for x E X+, I(LnZ(t)xll is less than or equal to
On the other hand, the latter is less than or equal to
which is equal to
where the additivity of the norm on X+ is used. The change of variables
shows that
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274
so that, for arbitrary x E X,
IILn Z(t)xll is less than or equal to
Hence and consequently
Finally
r t1 (L) ::; r t1 [B(O+ - TO)-l] and this proves the claim. <> ProoJofTheorem 12.1: Since Uo(.) E
U(t)
=
Uo(t)
+
H~,
the Duhamel equation
lot U(s)BUo(t - s)ds ,
written abstractly as
U-LU= Uo , has a unique solution U E
H~
given by
because r t1 (L) ::; r t1 [B(O+ - TO)-l] < 1. Thus {U(t); t 2: O} is bounded. This ends the first part of the theorem. The second part is dealt with similarly by introducing
Uo(t) = Uo( -t) , U(t) = U( -t) (t 2: 0) and noting that
-
-
U+LU= Uo where
L: Z E H~ ~ lot Z(s)BUo(t -
One proves, as in Lemma 12.2, that
and we proceed as in the first part.
<>
s)ds.
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Chapter 12. Limiting absorption principles
12.3
On the wave operators
This section is devoted to the existence of the following strong limits
limt. . . . s limt . . . . s
+oo
U(t)Uo( -t)
+oo
Uo(-t)U(t)
s limt ....... -
oo
U(t)Uo(-t).
We begin with Theorem 12.2 If both B(O± - TO)-l exist and ifru [B(O+ - TO)-l] then s limt ....... +oo U(t)Uo( -t) exists.
< 1,
~
O} is
Proof: According to the first part of Theorem 12.1, {U(t); t bounded. We set
IIUII =
sup IIU(t)ll · t~O
From
U(t) = Uo(t)
+
lot U(s)BUo(t - s)ds
one sees that
U(t)Uo(-t)x
=
x+
lot U(s)BUo(-s)xds.
Let x E X+ . Then
I:
IIU(s)BUo( -s)xll ds
~ IIUII l~ IIBUo(-s)xll ds = 'IU"III~ BUo( -S)xdsll = "u"III~oo BUo(S)Xdsll = IIUIIII-B(O- -
TO)-lxll ·
Hence limt ....... +oo U(t)Uo( -t)x exists for all x E X+ and finally for all x E X since X = X+ - X+. 0 Symmetrically we have the following Theorem 12.3 If both B(O± - TO)-l exist and ifru [B(O_ - TO)-l]
then s
limt. . . . -oo U(t)Uo( -t) exists.
< 1,
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276
Pr0o.f According to the second part of Theorem 12.1, { f i ( t )t,
> 0 ) is
bounded ( 6 ( t )= U ( - t ) ) and
-
where Uo(t)= Uo(-t), i.e.
Hence
6(s)BUo(s)xds; t 2 0.
f i ( t ) ~ o ( t )= xx By choosing z E X+,
Thus s limt,-, U(t)Uo(-t) exists because of X = X+ - X+. 0 The analysis of s limt-.+ooUo(-t)U(t) requires a technical preliminary result.
Lemma 12.3 The following assertions are equivalent. (i) B(O+ - To)-' exists and r, [B(O+- To)-'] < 1. (iz) (0,oo) c p(T), (A-T)-' is positive ( A > 0 ) and B(O+ -T)-'exists. We omit the proof which is quite similar to that of Lemma 8.1. We observe however that we have no similar result for B(0- - T)-' because U ( t ) need not be positive for t < 0 and (A - T)-' fails to be positive for X negative (and large). We are now ready to show
Theorem 12.4 If B(O+ - T)-'exists and if r, [B(o+- To)-']
< 1, then
s lim Uo(-t)U(t) exists. t-+m Proof: According to Lemma 12.3, B(O+ - T)-' exist or, equivalently (see Lemma 12.1), rt
s lim BU(s)ds exists. t-+m Jo
277
Chapter 12. Limiting absorption principles Thus
U(t) = Uo(t)
+
lot Uo(t - s)BU(s)ds
yields
Uo( -t)U(t)x = x +
lot Uo( -s)BU(s)xds.
By choosing x E X+ and noting that U(s) is positive for s :::: 0,
I:
IIUo( -s)BU(s)xll ds
~ sups~o IlUo(s)11 J: IIBU(s)xll ds =
sups~o IlUo(s)IIIIJ: BU(s)XdSIl
~ sups~o IIUo(s)IIIIJ~ BU(s)Xdsll = sups~o
This ends the proof since X
12.4
IlUo(s)IIIIB(O+ - T)-lXII ·
= x+ - x+. <>
The similarity of To and T
We show how the existence of wave operators provides a useful tool to show the similarity of the generators of the groups under consideration. Theorem 12.5 Let both B(O± - TO)-l exist and ru [B(O± - TO)-l] Then the following wave operators exist
W+(To, T)
:= s limt-++oo
Uo(-t)U(t)
W+(T, To) = s limt-++oo U( -t)Uo(t). Moreover,
and
<
1.
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278
Proof: The existence of W+(To, T) and W+(T, To) is contained in Theorems 12.3 and 12.4. We consider the identity [Uo( -t)U(t)] [U( -t)Uo(t)] = [U( -t)Uo(t)] [Uo( -t)U(t)] = I. Letting t
--+
+00 gives (12.8). On the other hand, for all s
~
0,
s limt-++oo U(s)U( -t)Uo(t)Uo( -s) = U(s)W+(T, To)Uo( -s)
= s limt-++oo U( -(t - s))Uo(t - s) = s limu-++oo U( -u)Uo(u)
=
W+(T, To).
Thus
U(s)W+(T, To) = W+(T, To)Uo(s).
(12.9)
The Laplace transform of (12.9) yields
(,\ - T)-lW+(T, To) = W+(T, To)('\ - TO)-l. Hence and
T = W+(T, To)To [W+(T, TO)]-l and the proof is complete. <) Before applying this theory to transport operators we complement the previous results by some converse theorems.
12.5
Converse results
In this section, we show the equivalence of various assertions used previously. The first result deals with CO-semigroups. Let {Uo(t); t ~ o} be a bounded positive CO-semigroup with generator To and {U(t); t ~ o} be the positive CO-semigroup generated by T = To + B where B E L+(X). According to Lemma 12.3, if B(O+ - T)-l exists then B(O+ - TO)-l exists and ru [B(O+ - TO)-l] < 1, and consequently, in view of Theorem 12.1, {U(t); t ~ O} is bounded. A simpler proof of this result as well as a converse statement are given in the following
c p(T) and B(O+-T)-l exists then {U(t); t ~ O} bounded. Conversely, if there exists c > 0 such that IlUo(t)xll ~ cIlxll for all x E X+ and if {U(t); t ~ O} is bounded, then ((0, +00) C p(T) and) B(O+ - T)-l exists.
Theorem 12.6 If (0, +00)
is
Chapter 12. Limiting absorption principles Proof: From the Duharnel equation
+
U ( t )= Uo(t)
1 t
Uo(t - s )BU(s)ds
we derive, for x E X+, the estimates
>
Hence {U(t);t 0 ) is bounded in view of X = X+ - X+. Conversely, i f { U ( t ) ;t > 0 ) is bounded then, for x E X+,
Hence
This implies, by the monotone convergence theorem, that s lirn
Jt
t-++a 0
BU(s)ds exists
which amounts t o the existence of B(O+ - T ) - l , in view of Lemma 12.1.
0
Corollary 12.1 Let there exist c > 0 such that IIUo(t)x11 2 c 11x11 for all x E X+. Then {U(t);t 0 ) is bounded if and only zf (0,+m) c P ( T ) and B(O+ - T ) - l exists.
>
Corollary 12.2 Let there exist c
> 0 such that IIUo(t)xII > ~11x11for all
x E X+. Then the perturbed semigroup { U ( t ) ;t
>
0 ) is bounded zf and only if B(O+ - To)-lexists and r , [B(o+- To)-'] < 1.
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280
Remark 12.1 If {U(t); t E R) is a positive and bounded co-group then there exists c > 0 such that IIUo(t)xll 2 c 11x11 for all x E X and all t E R. We assume, from now on, that {U(t); t E R) is a positive and bounded CO-group.
Theorem 12.7 The following assertions are equivalent: (i) s limt,+, Uo(-t)U (t) exists (ii) {Uo(-t) U(t) ;t > 0) is bounded (iii) {U(t); t 2 0) is bounded. Proof: We note that (i)+(ii) by the uniform boundedness theorem and (ii)==+(iii) because {U(t); t E R} is bounded. Let (zii) be true. Then, according to Corollary 12.2, B(O+ -To)-' exists and r, [B(o+ - TO)-'] < 1, and therefore the strong limit slimt,+, Uo(-t)U(t) exists in view of Theorem 12.4. 0 We end this section with
Theorem 12.8 The following assertions are equivalent: (i) s limt,+, U(t)Uo(-t) exists (ii) {U(t)Uo(-t) ; t > 0) is bounded (iii) {U(t); t 2 0) is bounded. Proof: The parts (i)& (ii) and (ii)* (zii) are clear. Let (iii) be true. Then, according to Corollary 12.2, B(O+ - To)-' exists and
On the other hand,
implies
and, since U(t) 2 Uo(t) (t rt
Let x E X+, then
> O),
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Chapter 12. Limiting absorption principles Hence, using the property
IlUo(t)xll
~
c IIxll for all x E X and all
t E R,
::::: J:UUo(S)BUo(-s)xlldS ::::: sUPr2:0
IIU(r)Uo( -r)llllxll·
By the monotone convergence theorem, S
lim t--->+oo
Jot
BUo( -s)ds exists,
i.e., according to Lemma 12.1,
B(O_ - T)-l exists. Finally s limt--->+oo U(t)Uo( -t) exists, in view of Theorem 12.2. <)
12.6
Scattering theory for transport operators
Let dJ-L be a Radon measure on R!' (n
~
1) with support V and let
0'(.,.) E L'f(Rn x V; dx ® dJ-L(v)) . Let {Uo(t); t E R} be the eo-group on X
Uo(t) : 1j;
E Ll(Rn x
V)
-+
= Ll(Rn
(12.10)
x V; dx ® dJ-L(v))
e - Jo' a(x-sv,v)ds1j;(x - tv, v)
with generator
To1j; = -v . ~ - O'(x, v)1j;(x, v) , 1j; { D(To) = {1j;
E £l(Rn x
The collision operator is defined by
V); v.ft;
E
D(To)
E Ll(Rn x
V)} .
Topics in Neutron Transport Theory
282 where
b{x,v,v') ~ 0 a.e. on R n x V x V
and h{ ., .) = [b{., v, .)dJ-L{v) E L';{R n x V; dx ® dJ-L{v)).
(12.11)
It is not difficult to see that (12.11) is necessary and sufficient in order for the operator B to be bounded. We denote by {U{t); t E R} the CO-group
with generator T=To+B .
We introduce some assumptions we will need later
+= 1 += sup
ess sup
(x ,v)
ess
(x,v)
u{x
+ sv, v)ds =
M+{u)
< +00
(12.12)
h{x
+ sv, v)ds =
M+{h)
< +00
(12.13)
= M_{h) < +00.
(12.14)
0
1 0
ess sup r+= h{x - sv, v)ds (x,v)
io
We observe that (12.12) is necessary and sufficient in order for {Uo{t); t :::; O} to be bounded, while {Uo{t); t ~ O} is always bounded in view of (12.1O). We are going to show that the assumptions above are exactly what is needed to define the wave operators. Theorem 12.9 (i) Let (12.13) be satisfied, then B{O+ - TO)-l exists and
(12.15) Conversely, if B{O+ - TO)-l exists and if (12.12) holds, then (12.13) also holds. (ii) Let (12.12) holds. Then B{O_ - TO)-l exists if and only if (12.14) holds, and then
Proof: We note that
{A - TO)-l'I/J =
1+=
e->'te -
J:
a(x-sv,v)ds 1jJ{x
- tv, v)dt (A> 0)
283
Chapter 12. Limiting absorption principles and
By choosing 'Ij;
~
0, the monotone convergence theorem yields
On the other hand,
J
dxd/1-(v)
J
00
b(x, v , v')d/1-(v') / e - Jo' u(x-sv' .V' )dS'Ij;(x - tv', v')dt
v
0
is equal to
J JJ 00
d/1-(v')
v
dt
0
h(x+tv' ,v')e- Jo'u(x+sv' .v')ds'Ij;(x,v')dx
(12.16)
Rn
which is less than or equal to
This shows that B(O+ - TO)-l exists and (12.15) holds under assumption (12.13) . Conversely, let B(O+-To)-l exist and let (12.12) hold, then (12 .16) shows that
(12.17) Thus, the left-hand side of (12.17) defines a continuous functional on
whence
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284
i.e. (12.13) holds. This ends the proof of the first part. To deal with the second part we observe that, under (12.12),
and we proceed as previously. <> The existence of wave operators for transport equations are direct consequences of the previous results. The results are summarized in the following Theorem 12.10 (i) Let (12.12) - (12.14) hold and let M+(h) < 1, then s limt-++oo U(t)Uo( -t) exists. (ii) Let (12.12) - (12.14) hold and let eM+(cr) M_(h) < 1, then the strong limit s limt--+-oo U(t)Uo( -t) exists. (iii) Let (12.13) hold and let M+(h) < 1, then s limt--++oo Uo( -t)U(t) exists.
This theorem takes advantage of the norms of the operators B(O± TO)-l while the relevant parameters are their spectral radii. We point out that B(O+ -TO)-l is positive and -B(O_ -TO)-l is also positive. To refine the above estimates we will assume, in the case where the spectral radius of B(O+ - TO)-l is positive (resp. the spectral radius of B(O_ - TO)-l is positive), that it is an isolated eigenvalue of B(O+ - TO)-l (resp. -B(O_TO)-l). This is true, for instance, if B(O± - TO)-l are power compact. Under this assumption we have Lemma 12.4 (i) Let (12.13) hold. We assume that
b+(v, v') = ess sup roo b(x + sv ' , v, v')ds xERn
Jo
is the kernel of a bounded operator B+ E L(Ll(V; dp,)). Then rcr [B(O+ - TO)-l] :S rcr(B+).
(ii) Let (12.12) and (12.14) hold. We assume that 00
I
L(v,v) = ess sup xERn
1
I
I
b(x - sv ,v,v )ds
0
is the kernel of a bounded operator B_ E L(Ll(V;dp,)). Then
285
Chapter 12. Limiting absorption principles Proof (i) Let ru [B(O+ - TO)-l] a corresponding eigenfunction
{
, , roo
Jvb(x,v,v)dJ.£(v)Jo
so that a1jJ(x,v):::;
i
e-
= a > 0 and
let 1jJ E L~(Rn x V) be
J.'oUx-sv,v ( ")d s1jJ(x-tv,v')dt=a1jJ(x,v) '
1+
00
b(x,v,v')dJ.£(v')
1jJ(x-tv',v')dt.
(12.18)
By integrating (12.18) with respect to x, we get acp(v):::;
i
dJ.£(v')
kn [1+
00
dx
b(x+tv',v,v')dt] 1jJ(x,v')
(12.19)
where cp(v) = (
JRn
1jJ(x,v)dx.
It follows from (12.19) that acp(v) :::;
i
b+(v, v')cp(v')dJ.£(v') = B+cp.
Hence ru(B+) ~ a = ru [B(O+ - TO)-l]
and this ends the proof of (i). To deal with (ii) we choose 1jJ E L~(Rn x V) such that where a = ru [-B(O_ - To)-~] > 0 and we proceed as in (i). 0 We complement Theorem 12.10 with Theorem 12.11 (i) Let (12.12) - (12.14) hold. If B(O+ - TO)-l is power compact and if ru(B+) < 1, then s limt-++oo U(t)Uo( -t) exists. (ii) Let (12.12) - (12.14) hold. If B(O_ - TO)-l is power compact and if eM+(u)ru(B_) < 1, then s lim t -+- oo U(t)Uo( -t) exists. (iii) Let (12.13) hold. If B(O+-To)-l is power compact and ifru(B+) < 1, then s limt-++oo Uo( -t)U(t) exists. Remark 12.2 Sufficient conditions ensuring the power compactness of B(O±TO)-l are given in [25]. In the context of Lebesgue measure (i.e. dJ.£(v) =
Ivl Iv'l
0 for ~ (and the cross-sections are compactly supported), then ru(B±) = 0 and therefore the wave operators exist regardless of the size (i.e. the norm) of B .
dv), if b(x,v,v')
=
286
12.7
Topics in Neutron nmsport Theory
Comments
The material in this chapter was taken from M. Mokhtar-Kharroubi 1251 where, more generally, unbounded (but To-bounded) collision operators B are considered. In the same spirit, an abstract two L1-spaces theory with applications to exterior problems (i.e. time asymptotic equivalence of the dynamics outside a bounded obstacle, with reflection at the boundary, and the free dynamics without obstacles) is given by M. Chabi, M. MokhtarKharroubi and P. Stefanov 191. We point out that the choice of L1spaces is a keypoint in this theory while positivity can be weakened (to some extent) by using domination arguments. However, for positive finite rank (or nuclear) perturbations, this theory was extended partially to general Banach lattices by M. Chabi and M. Mokhtar-Kharroubi [8]. The scattering theory for transport operators was initiated by J. Hejtmanek 1151, B. Simon [34] and developed by V. Protopopescu 1291, J. Voigt 1381, W. Schappacher [33] where Cook's method is used as well as positivity arguments. We mention a study of the range of wave operators by T. Umeda [36],by means of Enss decomposition principles. The wave operators for transport equations outside an obstacle (with reflection on the obstacle) as well as spectral theory are analyzed in P. Stefanov [35] (see also 191). Spectral results in the whole space are given in B. Montagnini 1261, J. Hejtmanek [16], A. Huber 1171, G. Greiner [14] and T. Umeda 1361. The Lax-Phillips formalism was applied to transport equations by H. Emamirad [lo] (see also H. Emarnirad [ll]).An abstract scattering theory in Banach lattices with applications to transport equations in L1 spaces was given by T. Umeda [37]. Nevertheless this theory does not apply, in general, to transport equations in P spaces (1 < p < co). We will show, in Chapter 13, that scattering theory for transport equations in P spaces (1 < p < co) is more suitably handled by Lin's factorization techniques [23]. Behind the existence of wave operators, in transport theory, there is the locally decaying property of transport operators which was pointed out by J. Voigt 1381. This property is tied to dispersive effects of velocity averages and will appear more transparently in Chapter 13. We point out a generalization of scattering theory to partly transparent surfaces (i.e. the dynamics in the whole space is subject to a transmission condition in some bounded region) by V. Protopopescu 1301. Useful relationships between the scattering operator and the albedo operator (for interior problems) are given in P. Arianfar and H. Emamirad [I],V. Protopopescu [31]and H .Emarnirad and V. Protopopescu 1121. The known literature on wave operators for transport equations is concerned only with bounded transport semigroups, referred to as subcritical or non-proliferating systems. This excludes, of course, the presence of point spectrum. At this point we mention an open (and probably difficult) problem : Assume that
Chapter 12. Limiting absorption principles
287
the cross-sections are compactly supported but the system is proliferating, giving rise to discrete point spectrum. Is it possible to define, in this case, generalized wave operators on a suitable subspace X I (with a reasonable description of X I ) ? The difficulty of the problem is tied, of course, to the lack of a reasonable functional calculus for transport operators. It is well known that the wave operators provide a mean to prove similarity results (see Theorem 12.5). This is known as the time dependent method of similarity. A stationary method (in Hilbert spaces) was given by T. Kato 1201 and extended (to Banach spaces) by S.C. Lin [24]. This method is applied to transport operators by M. Chabi [7]. At last, we point out that Runaway phenomena, in the kinetic theory of particle swarms, give rise to interesting scattering problems involving time dependent evolution problems (time dependent velocity) [2] - [6], [13]. It would be useful, and probably possible, to extend the abstract formalism of this chapter to time dependent evolution groups on L1(p) spaces. The aim of this chapter was merely to present one aspect of scattering theory (the existence of wave operators) tied to positivity in L1(p) spaces and motivated by transport theory. However, scattering theory is a very vast (and richer) subject for which we refer to the books [18] Chapter X, 1211, 1281 and the references therein.
References [I] P. Arianfar and H. Emarnirad. Relation between scattering and albedo operators in linear transport theory. Dansp. Theory Stat. Phys. 23(4) (1994) 517-531. [2] L. Arlotti. On the asymptotic behaviour of electrons in an ionized gas subject to time dependent electric field. Dansp. Theory Stat. Phys. 21(46) (1992) 733-752. [3] L. Arlotti and G. F'rosali. Long time behaviour of particle swarms in runaway regime. Op. theory: Adv. Appl. 51 131-143. Birkhauser Verlag, Basel, (1991). [4] L. Arlotti and G. F'rosali. Runaway particles for a Boltzmann-like transport equation. Math. Models Methods Appl. Sci. 2(2) (1992) 203-221. [5] G. Busoni and G. F'rosali. Asymptotic behaviour of a charged particle transport problem with time-varying acceleration field. Damp. Theory Stat. Phys. 21(46). (1992) 713-732.
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[6] G. Busoni and G. Frosali. Large time behaviour of drift velocity in a charged particle transport problem. Preprint. [7] M. Chabi. Similitude d'operateurs de transport. Methode stationnaire. Work in preparation. [8] M. Chabi and M. Mokhtar-Kharroubi. On perturbations of positive co-(semi)groups on Banach lattices and applications. J. Math. Anal. Appl. 202 (1996) 843-861. [9] M. Chabi, M. Mokhtar-Kharroubi and P. Stefanov. Scattering theory with two L1 spaces: application to transport equations with obstacles. Ann. Fac . Sci. Toulouse. (1997). To appear. [10] H. Emamirad. On the Lax and Phillips scattering theory for transport equation. J. Funct. Anal. 62 (1985) 276-303. [11] H. Emamirad. Scattering theory for linearized Boltzmann equation. Transp. Theory Stat. Phys. 16 (1987) 503-528. [12] H. Emamirad and V. Protopopescu. Relationship between the albedo and scattering operators for the Boltzmann equation with semitransparent boundary conditions. Math. Methods Appl. Sci. [13] G. Frosali, C.Van der Mee and S.L. Paveri Fontana. Conditions for runaway phenomena in the kinetic theory of particles swarms. J. Math. Phys. 30(5) (1989) 1177-1186. [14] G. Greiner. Spectral properties and asymptotic behavior of the linear transport equation. Math. Z. 185 (1984) 167-177. [15] J . Hejtmanek. Scattering theory of the linear Boltzmann operator. Comm. Math. Phys. 43 (1975) 109-120. [16] J. Hejtmanek. Dynamics and spectrum of the linear multiple scattering operator in the Banach lattice L1(R3 x R 3). Transp . Theory Stat. Phys. 8(1) (1979) 29-44. [17] A. Huber. Spectral properties of the linear multiple scattering operator in L1-Banach lattices. Int. Eq. Op. Theory 6 (1983) 357-371. [18] T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag, 1984. [19] T . Kato. Scattering theory. Studies in mathematics. Math. Assoc. Amer. 7 (1971) 90-115.
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[20] T. Kato. Wave operators and similarity for some non-self-adjoint operators. Math. Ann. 162 (1966) 258-279. [21] P.D. Lax and RS. Phillips. Scattering Theory. Academic Press, Inc, 1989. [22] P.D. Lax and RS. Phillips. Scattering theory for dissipative systems. J. Funct. Anal. 14 (1973) 172-235. [23] S.C. Lin. Wave operators and similarity for generators of semigroups in Banach spaces. Trans. Amer. Math. Soc. 139 (1969) 469-494. [24] S.C. Lin. A stationary approach to perturbation of operators in Banach spaces. J. Math. Anal. Appl. 32 (1970) 352-369. [25] M. Mokhtar-Kharroubi. Limiting absorption principles and wave operators on L1(J.-L) spaces. Applications to transport theory. J. Funct. Anal. 115 N01 (1993) 119-145. [26] B. Montagnini. The eigenvalue spectrum of the linear Boltzmann operator in Ll(Rn) and L2(Rn) . Meccanica. (1979) 134-144. [27] R Nagel (Ed). One-parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, 1184, Springer-Verlag, 1986. [28] V. Petkov. Scattering Theory for Hyperbolic Operators. North-Holland, 1989. [29] V. Protopopescu. On the scattering matrix for the linear Boltzmann equation. Rev. Roum. Phys. 21 (1976) 991-994. [30] V. Protopopescu. Une generalisation de la theorie de scattering pour les equations du trarISport lineaires. C. R . Acad. Sc. Serie 1. 317 (1993) 1191-1196. [31] V. Protopopescu. Relation entre les operateurs d'albedo et de scattering avec des conditions aux frontieres non transparentes. C. R. Acad. Sc. Serie 1. 318 (1994) 83-86. [32] M. Reed and B. Simon. Methods of Modern Mathematical Physics. Vol 3: Scattering Theory. Academic Press, 1979. [33] W. Schappacher. Scattering theory for the linear Boltzmann equation. Ber Math. Stat Sekt. Forschungszentrum. Graz. 69 (1976) . [34] B. Simon. Existence of the scattering matrix for the linearized Boltzmann equation. Comm. Math. Phys. 41 (1975) 99-108.
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[35J P. Stefanov. Spectral and scattering theory for the linear Boltzmann equation in Exterior domain. Math. Nachr. 137 (1988) 63-77. [36J T. Umeda. Scattering and spectral theory for the linear Boltzmann equation. J. Math. Kyoto Univ. 24(2) (1984) 205-218. [37J T. Umeda. Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators. J. Math . Soc. Japan 38(4) (1986) 617-625. [38J J. Voigt. On the existence of the scattering operator for the linearized Boltzmann equation. J. Math. Anal. Appl. 58 (1977) 541-558.
Chapter 13
Lin's factorization formalism and applications to transport theory 13.1
Introduction
This chapter is devoted to scattering theory for transport operators in LP spaces (1 < p < (0). The formalism presented in Chapter 12 is based, in a crucial way, on a characteristic property of the L1 norm, namely the additivity on the positive cone. We point out the existence of a formalism in general Banach lattices by T. Umeda [9]. However, in LP spaces (1 < p < (0), apart from the case where the velocity space is bounded away from zero, T . Umeda's assumptions are hardly compatible with natural assumptions on the cross-sections. An alternative approach is provided by factorization techniques used by many authors, particularly by T. Kato [4] and S.C. Lin [5] . We will present some of S.C. Lin's results [5] in reflexive Banach spaces and show how they can be used in the context of transport theory. As in the preceding chapter, we only deal with the existence of wave operators. Actually, S.C. Lin deals with unbounded perturbations and this makes the theory quite technical. We will restrict ourselves to bounded perturbations. This allows a considerable simplification of the different proofs and, moreover, some of S.C. Lin's assumptions are no longer necessary in this context. We consider the following framework. Let X be 291
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a reflexive Banach space and let {Uo(t)j t E R} be a co-group acting on X, with generator To. We denote by {U(t)j t E R} the co-group generated by T = To + B where B E L(X) . The general philosophy of the factorization technique is that the perturbation B be factorizable as B = B1B2 where B1 E L(WjX) , B2 E L(Xj W)
(for some auxiliary Banach space W) in such a way that B2 be "smooth" relative to {Uo(t)j t E R} and Bi be "smooth" relative to {UO'(t) j t E R} the dual group in the dual space X*. The concept of smoothness is explained below. As will be seen in the applications to transport theory, it is natural to allow W to be different from the initial space X . We will restrict ourselves to one wave operator, for example s lim U(t)Uo(-t). t-++oo
The other wave operators can be handled by exactly the same techniques. The smoothness assumptions we need for the analysis of this wave operator are the following. There exists p E ]1, oo[ such that, for all x E X and x* E X*,
1 0
-00
IIB2UO(s)xll~ ds < 00
r+oo
'Jo
IIB~Uo(s)x*II'. ds < 00
(13.1)
and
(13.2) where IIlIw is the norm in W, IIIIL(W) is the corresponding operator norm in L(Wj W) and p* is the conjugate exponent of p. We point out that the reflexivity assumption on X is intended to ensure that the dual group be strongly continuous. When dealing with applications to transport equations in V' spaces (1 < p < 00), we will see some interesting differences, with respect to the previous L1 theory, as regard to the relevant assumptions on the cross-sections and, in particular, the role of the velocity measure df.L(v) in the neighborhood of the origin is more transparent.
13.2
A preliminary result
We give a technical result related to Baire category theorem we need below. Let Y1 and Y2 be two Banach spaces and let H(Y1, Y2 ) := {Z: [0, oo[
--+
L(Y1' Y2 )j Z is strongly continuous} .
293
Chapter 13. Link factorization formalism For each q E [I,cm[ we define the vector space
For the reader's convenience we give a proof of the followingtechnical result. Lemma 13.1 For all Z E Hq(Yl, Y2)
Proof: Let n E N and
Clearly c, is a continuous half-norm in since {Z(t); t bounded. On the other hand, by assumption,
> 0) is locally
Thus c : Y E Yl
+
(LODllz(t)~ll;~)~
is a lower semicontinuous half-norm in Yl. Hence, for each n E N,
Mn := {y E Yl; ~ ( y I ) n) is closed and Yl = U ~ E N M ~ . It follows, from Baire's category theorem (see, for instance, [2] p. 15), that there exists an integer no such that M,, has a nonempty interior. Let D(yo,a) c Mn, be a closed ball with radius a centered at yo. Thus
The half-norm property yields sup C(Y)I a-1 YO) Ilvlly, 51
+ no] < +oo
which ends the proof. 0 We endow Hq(Yl, &) with the norm (13.3). We point out that Hq(Yl, 5) is not complete. We denote by Hq(y1, 5)its completion.
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13.3
On the wave operator s limt-++oo U(t)Uo( -t)
Yet some preliminary results are necessary. Let p E J1, 00[ . We introduce
where p* is the conjugate exponent of p and
Q: Z E Hp.(W*)
---->
1t B;Uo(t - s)B2Z(s)ds.
Clearly QZ is strongly continuous. Some properties of Q are given in Lemma 13.2 Let (13.2) be satisfied. Then
Q
E
r
CXJ
L(Hp• (W*); Hp. (W*)) and IIQII :S Jo
IIB 2 UO(s)Bl IIL(W) ds .
Denoting still by Q the unique continuous extension to H p. (W*) we have
Proof Let Z E Hp. (W*) and x· IIQZ(t)x*II' . is less than or equal to ~
t
[1 IIB;Uo (t-S)B 2 11 dS ] Let
1
W*.
By Holder's inequality,
t
11IB;uo(t-s)B21111Z(s)x*II'.ds.
CXJ
f =
Then
p
E
1
IIB;Uo (s)B211 ds.
CXJ
.I.1
CXJ
IIQZ(t)x*II'. dt :S f2(,
IIZ(s)x*II'. ds
whence
IIQZII :Sf· IIZII· This proves the first claim. To prove that QZ E Hp.(W*) even if Z E H p. (W*) we argue as follows . Let {Zn} C Hp. (W*) such that Zn ----> Z in H p. (W*) . Let t > 0 be fixed arbitrarily. Then
IIQZn(t)x· - QZrn(t)x*IIw·
:S
J:
IIBiUO'(t - S)B211 IIZn(s)x· - Zrn(s)x*IIw. ds
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Chapter 13. Lin 's factorization formalism and
SUPtE[O,IjIIQZn(t)X· - QZm(t)x*llw·
~
(I:
IIBiUO'(s)B:iII P
ds)*(/~ IIZn(s)x* -
Zm(s)x*II'. ds)';' .
Hence {QZn(t)X* ;t E [0, 't]} nEN is a Cauchy sequence in C( [0, t] ; W*) and this shows that Q Z is strongly continuous. <> The following result shows that the perturbed group {U(t) ;t E R} inherits the smoothness properties of {Uo(t) ;t E R} . Lemma 13.3 Let (13.1) and (13.2) be satisfied and let rO"(Q) < 1. Then
10
00
IIBiU*(s)x*II'. ds <
00
V x* E X *.
Proof The Duhamel equation
U*(t) = U;(t) + lot U;(t - s)B*U*(s)ds shows that Z(t)
= BiU*(t)
satisfies
Z(t) = BiU;(t) + lot BiU;(t - s)B;Z(s)ds .
(13.4)
In view of the second part of (13.1), BiUO' E Hp.(W*). Writing (13.4)
abstractly as (13.5) one sees that (13.5) has a unique solution in Hp.(W*) since rO"(Q) < 1. In view of Lemma 13.2, this solution belongs to Hp. (W*) . Finally it coincides with BiU* by a Gronwall argument. Hence BiU* E Hp. (W*) and this proves the claim. <> We are in a position to prove the main result of this section. Theorem 13.1 Let (13.1) and (13.2) be satisfied and let
r+
Jo
oo
IIB 2 UO(s)B 1 1I L (w) ds < 1.
Then s limt~+oo U(t)Uo(-t) exists.
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Topics in Neutron Transport Theory Proof From the Duhamel relation
U(t) = Uo(t) +
lot U(s)BUo(t -
s)ds
we obtain, for x E X and x* E X* ,
(U(t)Uo(-t)x,x*)
= (x,x*) + J: (U(s)BUo(-s)x,x*)ds = (x, x*)
+ J: (B 2Uo( -s)x, BiU*(s)x*)ds
where (.,.) is the duality pairing between X and X*. According to (13.1),
looo IIB U 2
O(
-s )xllev ds < 00
V x E X.
On the other hand, by (13.1) and Lemma 13.3,
looo IIBiU*(s)x*II~" ds <
00
V x* E X* .
Hence
~ (J~ IIB2UO(-s)xllevds)i(J~ IIBiU*(s)x*II~" dS)?
<
00.
It follows that lim U(t)Uo( -t)x exists weakly. t-.+oo We denote by Gx this (weak) limit. We note that
(Gx - U(t)Uo( -t)x, x*) =
J~ (B 2UO(-s)x, BiU*(s)x*)ds
~ [(J~ IIB2UO(-s)xllev dS)*] [(J~ IIBiU*(s)x*II~" dS)? ] whence
IIGx - U(t)Uo(-t)xll
= sUPllx"11:51 (Gx
- U(t)Uo( -t)x, x*)
~ SUPllx"11:51 [(J~ IIBiU*(s)x*II~" dS)?] [(J~ II B 2Uo(-s)xllev ds)* 1·
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297
In view of Lemma 13.1 and Lemma 13.3, sup [( roo IIx-1I9
Jo
IIBrU*(s)x*II'_ dS)~] < 00.
Therefore lim IIGx - U(t)Uo( -t)xll t-++oo and the proof is complete.
13.4
-+
0
0
Application to transport groups. Part 1
We devote the rest of this chapter to applications of this formalism to transport equations. The main point is to find out na.t ural factorizations of the collision operator satisfying the requirements of the abstract theory. We will present two such factorizations. We point out that this approach does not rely to any positivity assumption. We assume, for the sake of simplicity, that the collision frequency is non-negative. This section is devoted to the first factorization. We recall the general setting we consider. Let dJ.l be a Radon measure on R n (n ~ 1) with support V and let
a(.,.) E L'+(Rn x V; dx ® dJ.l(v)) { sUP(x,v)ERnxV
J::
(13.6)
a(x - sv, v)ds <
00.
Let 1 < p < 00 and let X = V(Rn x V; dx ® dJ.l(v)). We denote by E R} the CO-group on X
{Uo(t); t
Uo(t) : '1jJ E LP(Rn x V)
-+
e- Jot CT{x-sv,v)ds'I/J(x_ tv, v)
with generator
To'I/J = -v.~ - a(x, v)'1jJ(x, v) ; 'I/J E D(To) { D(To) = {'I/J
E
V(Rn x V); v.~ E V(Rn x V) } .
The collision operator, defined by
B: 'I/J E LP(Rn x V)
-+
Iv b(x,v,v')'I/J(x,v')dJ.l(v') ,
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is assumed to be bounded on LP(Rn x V) . In this section, the precise assumption is hl(" ' ):= j)b( .,v;.)ldtt(V) ELoo(Rn x V)'
(13,7) {
h2("' ):= j V Ib(., ., v')1 dtt(v ' ) E
LOO(~ x V).
Let
T(X, v, v)
= {
Hence
b(x, v, v ) I
'f b(
b x,v ,v Ib(x ,v ,v ' )
I
1
X , V, V
')
..i.
-r
0
oif b(x, V, v') = O. = Ib(x, v, v ) IT(X, v, V ). I
I
One sees that where B2: LP(Rn x V) {
'If; (x, v')
-t
-t
~(Rn x
V x V)
Ib(x,v,v' )lp T(X,V, v')'If;(x, v')
and B 1 : LP(Rn x V x V) { G(x,v,v' )
-t
-t
LP(Rn x V)
j v Ib(x,v,V,)l pl. G(x,v,v')dtt(v' )
:*
where ~ + = 1. It is easy to see, in view of (13.7), that Bl and B2 are bounded operators
Thus
x
= LP(Rn x V) , W = LP(Rn x V x V).
We observe that
•
Uo(t) : 'If; E LP (Rn x V)
-t
r' u(x-sv''If;(x v)ds + tv, v).
eJo
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Chapter 13. Lin's factorization formalism Moreover,
B 2 : VO (Rn x V x V) { cP
-+
J
-+
VO (Rn x V)
i
v Ib(x, v, v')l T(X, v, v')cp(x, v, v')dll(V)
and
Bi: vO(Rn x V) { cp
-+
E
VO (Rn x V)
VO(R n x V x V) (13.8)
---->
Ib(x,v,v') lplO cp(x,v) E vO(Rn x V x V).
We are ready to prove Proposition 13.1 Besides (13.6) and (13.7), we assume that hI and h2 satisfy the conditions
1 0
sup (x,v')ERnxV
hl(x+tv',v')dt <
00;
sup (X ,V' )ERnxV
-00
roo h (x-tv',v')dt < 2
Jo
00
then, for all 'IjJ E X and 'IjJ* E X*,
Proof Let 'IjJ E V(Rn x V). Then
+OO
and, in view of (13.6), there exists c > 0 (c = sup(x,v') e J such that
-00
= cP
J
Rn x V
'
P
I
I
<1(x-sv ,v )ds)
hl(X, v) I'IjJ(x - tv "I , v) dxdll(V) ,
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Hence
This shows the first part of the claim. Let 'Ij;* E If (Rn x V) then, according to (13.8) , ...L
ft
B;U;(t)'Ij;* = Ib(x, v, v')1 p. eJo u(x+sv,v)ds'lj;*(x
+ tv, v)
and therefore
Thus
and the proof is complete. 0 The conditions on a(., .), hI (., .) and h2 (., .) are natural in the context of scattering transport, as was shown in the preceding chapter, in Ll setting. In the case where a(x, v) and b(x, v, Vi) vanish outside a bounded subset n c R n , such conditions can be satisfied as follows. For instance, considering the collision frequency a(.,.) and introducing polar coordinates v = pw (Iwl = 1),
1
00
1
00
a(x - tpw ,v)dt = p-l
a(x - tw ,v)dt.
Therefore, since the length of {x - tw; t E R} n the diameter d of n, sup (x,v)ER" xV
roo a(x-tv,v)dt~d
Jo
n is
less than or equal to
sup a(x,v) (x,v)EnxV Ivl
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Chapter 13. Lin's factorization formalism
Similarly, the conditions on hi (i = 1,2) are satisfied if sUP(x,V)EflxV hif~() is finite . Let us consider the more intricate condition (13.9)
We restrict ourselves to compactly supported cross-sections. Let G E W := £p(Rn x V x V). Then
= b(x, v, v)' I T(X, v, v)e .1 p
I
B2UO(s)B1 G
x J v Ib(x - sv', v', v")
I
pl.
I
-
1" u(x-sv v ,)ds 0
'
G(x - sv', v', v")df.L(v")
so that
II B 2Uo(s)B1Gllev
J J
~ cP x
dxdf.L(v)df.L(v' ) Ib(x, v, v')1
[J
...I!...
v Ib(x - sv', v', v")1 df.L(V")]
p.
P v IG(x - sv', v', v")I df.L(v")
=cpJ dxdf.L(v')h1(x,v') [h2(X-SV',v')]?" JvlG(x-sv',v',v")IP df.L(v") =cpJ dxdf.L(v')h1(x+sv',v') [h2(x,v')]?" JvlG(x,v' ,v")IP df.L(v"). Hence
IIB2UO(s)B11IL(w)
.!
~c
,sup
...!..
[h1(x+sv',v')]p [h 2(x ,v')]p·
(x,v)ERn x V
Since b(x , v, v') = 0 for x ¢.
n, then the
constraints
x En, x + sv' E n imply
IIB2UO(s)B11I L(w)
~c
.!
sup XEfl,lv'I::;~
...!..
[h1(x+sv', v')]p [h 2(x,v')]p·
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302
where d is the diameter of n. It is then natural to introduce the functions
and hl(S):= sup h 1 (v') , h 2(s):= sup h2(V'). Iv'I~!
(13.10)
Iv'I~!
Finally, we have proved the following result. Proposition 13.2 Let the cross-sections be supported by a bounded set n c Rn with diameter d. Then the condition (13.9) is satisfied provided that the function (13.11) is integrable at infinity. Proof. Clearly, we only need the integrability condition (13.9) at infinity. The proof is complete, in view of the previous calculations. 0 We point out that if the velocity space is bounded away from zero and if the cross-sections are compactly supported, then all the other conditions on the cross-sections introduced in this section are trivially satisfied. We give, now, a sufficient condition on the scattering kernel ensuring the condition of Proposition 13.2 when the velocity space is not bounded away from zero.
Corollary 13.1 Let h 1 (v) = O(lvla) andh2(v) = O(lvl.B) in the neighborhood of the origin, with ~ + > 1, then the condition in Proposition 13.2 is satisfied.
f.
Proof. It is a simple consequence of the definitions (13.10).
13.5
0
Application to transport groups. Part 2
We consider now another natural factorization suggested by the physical models of neutron transport. We recall that the collision frequency is given by a(x, v) = Ivl [aa(x, v) + as(x, v)] where aa(., .) and a s (., .) are, respectively, the macroscopic cross-sections for absorption and scattering, while the scattering kernel has the form b(x,v,v') = {3(x,v,v') Iv' l as(x,v') .
303
Chapter 13. Lin's factorization formalism Thus, we consider here the case
inducing the natural factorization
where and
Bl : cp E p(Rn x V)
-t
Iv
b1(x, V, V')cp(X, v')dJ1.(v'}.
Here, the factorization of collision operator takes place in X = p(Rn x V) (i.e. W = X). We assume, for the sequel, that b2 is bounded and that
b2(x,v')=b 1 (x,v,v')=Oifxrt.n where n c Rn is bounded. The following assumptions will appear naturally in the sequel
[1
sup (x ,v)ERn xV
h2(" '):=
0
+ tv, vW dt] < 00
Ib2(x
(13.12)
-00
J Ib V
1( ., .,
v')1 dJ1.(v' ) (13.13)
{
sup(x,v')ERn xV
h 1 (., .) :=
Iv Ib
1 ( .,
[J~ h2(X -
tv', v')dt] <
00
v , ·)1 dJ1.(v) E LOO(Rn x V) .
(13.14)
Proposition 13.3 Let (13.12) - (13.14) be satisfied. Then
l~ IIB2UO(t)cpIIP dt <
1+
00
00
cp EX,
IIBiU;(t)cp*IIP' dt < 00
cp" E X* .
Proof. Let cp E p(Rn x V). We have
IIB2UO(t)cpIIP
=
J
Rn xV
Ib 2(X, v)e -
Jo' <7(x-sv,v)dscp(x -
P tv, v)I dxdJ1.(v)
Topics in Neutron Transport Theory
304 +OO <7(x-sv,v)ds
where c = sup(x,v) e J
-00
,
so that
This proves the first claim. Let cp* E V· (Rn x V). Then
B;Uo(t)cp* =
Iv
b1 (x,v',v)eJo'<7(x+sv',V')dScp*(x+tv',v')dJ-L(v').
By using Holder's inequality and (13.14), we obtain
Hence
and this ends the proof in view of (13.13). We consider now the condition (13 .2)
<>
Proposition 13.4 We suppose that h2 E Loo(Rn x V) . Then, there exists a constant c such that 1.
IIB2UO(t)Blll
~ c (x,v') sup [r dJ-L(v) Ib1(X, v, v')I] " Jlvl<5.~
where d is the diameter of fl. Proof Let cp E V(Rn x V). We have
(t
> 0)
Chapter 13. Link factorization formalism Using the fact that t 2 0 and u ( . ,.)
> 0,
IB2U0(t)BlcplP
Jv Jbl(x- tv1v,v1)I I~ ( x - t v , ~
5 lb2(xl~)lp llh~llt Hence
and consequently P
I I ~ ~ ~ ~ ( t ) BI l cIlh2lltpl]~
SUP
(x,.')
[/V
C(V) Ib2(x + t v l ~ ) l~P b l ( x ~ v ~IIvIIp ~)l]
Thus
Finally, using the fact that bz(z, v) = bl(z, v, v') = 0 if z
6 R,
4 and therefore E = 11 b2 1ILm 11 h2 llLm is the desired constant. In particular we have the following
0
Corollary 13.2 Let bl(., ., .) E L*(Rn x V x V) and assume that
is integrable at infinity. Then
11 B2UO(t)B111 dt < 00.
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306
Remark 13.1 When dJL(v) = dv (the Lebesgue measure) the integmbility of IIB2UO(t)Blll at infinity is ensured if n > p. To deal with the case n :S p it is necessary to impose conditions on b1 (x, v, v') at small v. We leave the details to the reader. Finally, in the case where the assumption of Corollary 13.2 holds, the estimate
indicates the mnge of the different pammeters ensuring the important conoo dition Jo IIB 2 UO(t)Blll dt < 1. In particular, this assumption is fulfilled if the diameter of n is small enough.
13.6
Comments
The material in this chapter was taken from M. Mokhtar-Kharroubi [6J. S.C. Lin's theory [5J was extended to non-reflexive Banach spaces by D.E. Evans [3J. The factorization formalism described in this chapter extends without difficulties to countable factorizations [6], i.e. to perturbations (13.16) where B2' E L(Xj wm), BI" E L(wmj X) , wm (m E N) are auxiliary Banach spaces and the series (13.16) is normally convergent, i.e. 00
L
m=l
IIBI"IIIIB2'11 < 00 .
This countable factorization could have, in principle, applications to transport equations. Indeed, if the kernel of the collision operator B, i.e. the scattering kernel, can be expanded in the form 00
b(x, v, v')
=
L
ff'(x, v)f;'(x, v)
m=l
00
L
m=l
Ilff'llllf;'11 < 00
(13.17)
Chapter 13. Lin's factorization formalism
307
wm
then the countable factorization formalism could be applicable with = LP(Rn). The condition (13 .17) expresses a kind of nuclearity ofthe collision operator B with respect to velocities. This is not, a priori, an artificial assumption since B is an integral operator with respect to velocities. However this approach is somewhat abstract since the functions fr, f2' (m E N) are unknown and it is not possible to translate, in terms of conditions on b( ., ., .), the different assumptions (besides (13.17)) we need on fi, f2' (m EN). Let us comment briefly on the assumptions (13.1) and (13.2), in the context of transport operators, say, of Section 13.5. Since b1 (., v , v') and b2 (. , v') are supported by c Rn bounded, then IIB2 UO(t)cpll and IIBiUo(t)cp*11 involve, in fact, local (in x) norms and as such they go to zero as t -> ±oo. This is the locally decaying property of free transport groups (see J. Voigt [10]). However, (13.1) imposes a sufficiently fast decaying in order to reach the integrability condition. The condition (13 .2) is yet much stronger since it involves a uniform (with respect to cp) integrability. One can interpret this better time decaying as an effect of velocity averages or, more precisely, the behaviour of the velocity measure dp,( v) at small velocities. This dispersive effect of velocity averages was already exploited, in the context of non-linear kinetic equations, by C. Bardos and P. Degond [1]. We refer to B. Perthame [7] and references therein, for recent developments on dispersive effects for kinetic equations.
n
References [1] C. Bardos and P. Degond. Global existence for Vlasov-Poisson equation in 3 space variables with small initial data. Ann. Inst Henri Poincare. Anal Non Lineaire 2 (1985) 101-118. [2] H. Brezis. Analyse Fonctionnelle: Theorie et Applications. Masson, Paris, 1983. [3] D .E. Evans. Smooth perturbations in non-reflexive Banach spaces. Math. Ann. 221 (1976) 183-194. [4] T . Kato. Wave operators and similarity for some non-self-adjoint operators. Math. Ann. 162 (1966) 258-279. [5] S.C. Lin. Wave operators and similarity for generators of semigroups in Banach spaces. Trans. Amer. Math. Soc. 139 (1969) 469-494. [6] M. Mokhtar-Kharroubi. Scattering problems in transport theory. Talk given in University of Tubingen, Germany, June 1994. Unpublished.
308
Topics in Neutron Transport Theory
[7J B. Perthame. Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Part. Diff. Eq. (1996). [8J W. Greenberg, C. Van der Mee and V. Protopopescu. Boundary Value Problems in Abstract Kinetic Theory. Birkhauser Verlag, 1987. [9J T. Umeda. Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators. J. Math. Soc. Japan 38(4) (1986) 617-625. [lOJ J . Voigt. On the existence of the scattering operator for the linearized Boltzmann equation. J. Math. Anal. Appl. 58 (1977) 541-558.
Chapter 14
Inverse scattering and albedo operator. By M. Choulli and P. Stefanov
14.1
Introduction
Consider the Boltzmann equation
~~ = -v·\7xu(t,x,v)-aa(x,v)u(t,x,v)+
i
k(X,VI,V)U(t,x,VI)dv' (14.1)
in Rn x V 3 (x, v), V being an open subset of Rn , n 2: 2. Equation (14.1) describes the dynamics of a flow of particles in R n under the assumption that the interaction between them is negligible (no non-linear terms). This is the case for example for a low-density flow of neutrons. The term involving aa describes the loss of particles from (x, v) ERn x V due to absorption or scattering into another point (x, Vi), while the last term in (14.1) involving k represents the production at x E Rn of particles with velocity v form particles with velocity Vi . The total rate of this production at (x, Vi) is given by
ap(x, Vi) =
i
k(x,v',v)dv.
Following [14] we say that the pair (aa, k) is admissible, if (i) O:S aa E Loo(R n x V), (ii) O:S k(x, Vi,.) E Ll(V) a.e. (x, Vi) ERn x V and a p E Loo(Rn x V)
309
Topics in Neutron Transport Theory
310
(iii) There is an open bounded set X eRn, such that k(x,v',v) and
Ga(x, v) vanish if x
f/. x.
Denote To = -v · \7 x with domain D(To) = {J E p(Rn x V); V · \7 x f E LI(Rn x V)} . It is well-known that To is a generator of a strongly continuous group Uo(t)f = f(x - tv, v) of isometries on LI(Rn x V) preserving the non-negative functions. Following the widely accepted notations, let us introduce the operators
-Ga(x, v)f(x, v), [A 2f](x, v)
=
J
v k(x, v', v)f(x, v') dv' , T = To + Al
+ A2 =
TI
+ A2
and set A = Al + A 2. Operators Al and A2 are bounded on P (R n x V) and T I , T are generators of strongly continuous groups UI(t) = etT1 , U(t) = etT , respectively [14]. For UI(t) we have an explicit formula
[UI(t)f](x , v) = e -
Jo'
C7
a
(x-sv,v)ds
f(x - tv, v),
(14.2)
while for U (t) we have (14.3) We work in the Banach space LI(Rn x V), so here IIU(t)11 is the operator norm of U(t) in £(p(Rn x V)). It should be mentioned also that U(t) preserves the cone of non-negative functions for t :::: O. One can define the wave operators associated with T, To by s-lim U(t)Uo(-t),
(14.4)
s-lim Uo( -t)U(t) .
(14.5)
t-+cx>
t-+cx>
If W _,
W+ exist,
then one can define the scattering operator
S=W+W_ as a bounded operator in LI(Rn x V). Scattering theory for (14.1) has been developed in [8], [15], [21] and we refer to these papers (see also [14]) for sufficient conditions guaranteeing the existence of S . We would like to mention here also [11], [19], [6], [16], [22] . An abstract approach based on the Limiting Absorption Principle has been proposed in [10]. We will show in Section 14.2 however that S can always be defined as an operator S : L~(Rn x V\ {O}) -> L~oc(Rn x V\ {O}). The first inverse problem we are interested in is the following: Does S determine uniquely G a , k? We show that the answer is affirmative if G a is independent of v.
Chapter 14. Inverse scattering and albedo operator
311
Theorem 14.1 Let (O'a, k), (u a , k) be two admissible pairs such that O'a, u a do not depend on v and denote by S, S the corresponding scattering operators. Then, if S = S, we have O'a = Ua , k = k.
One can relax a little bit the assumption that O'a, u a do not depend on v . For example, assume that V is spherically symmetric and O'a = O'a(x, Ivl), u a = ua(x, Ivl). Then it is clear from the proof (see also (14.6) below) that the uniqueness result in Theorem 14.1 still holds. However, it is important to note that Theorem 14.1 fails to be true for general O'a. Consider for example the pairs (O'a, 0), (ua , O) , where u a = O'a(x + p(x, v)v, v), with p some nontrivial continuous function such that p(x, v)v is bounded on Rn x V . Then, if (O'a , 0) is admissible, so is (u a , 0). Since k = 0, we have Sf = e -
J::= ua,{x-sv,v)ds f,
(k = 0)
(14.6)
and it is easy to see that S = S although O'a =I- u a. Note that if k = 0, and O'a does not depend on v, it follows from (14.6) that S determines uniquely the X-ray transform of O'a and therefore O'a. The proof of Theorem 14.1 is constructive, it implies an explicit procedure for recovering O'a and k from S. It turns out that all the information necessary to recover 0'a, k is contained in the behavior of the Schwartz kernel S(x,v,x' , v') of S near the singularities (x,v) = (x' , v') and x = x', v =I- v', respectively. Next object we consider is the so-called albedo operator $. Assume that X is convex and has CI-smooth boundary ax. We propose the following definition of $ which generalizes that given in [1], [7], [12]. Denote f ± = {(x , v) E ax x V; ±n(x) · v > O} , where n(x) is the outer normal to ax at x E ax. Consider the measure d1, = In(x) . v ldJ.£( x)dv on f ±, where dJ.£(x) is the measure on ax. Let us solve the problem
(at - T)u
=
0 in R x X x V,
Ult«o
=
0
(14.7)
for u(t,x , v) , where g E L~(R; £l(f_, d~)) and T is considered as a differential operator in X x V. We will see that (14.7) has a unique solution u E C(R; LI(X x V)) and one can define the albedo operator $ by (14.8)
Topics in Neutron Transport Theory
312
Operator $ : L~ (R; L1 (r _, d~)) - t L{oc (R; L1 (r +, dO) maps the incoming flux on ax to the outgoing flux on ax. It can be seen that $g can be defined more generally for 9 E L1 (R x r _, dt d~) with 9 = 0 for t «0. It has been shown in [1], [7], [12] that there is a relationship between Sand $. We show below that in fact $ determines S uniquely and conversely, S determines $ uniquely by means of explicit formulae. To this end, let us define the extension operators E± and the restriction (trace) operators R± as follows. Set D = {(x, v) ERn x V\ {O}; ::It E R, such that x - tv EX},
(14.9)
and define the functions T±(X, v)
= max{t E R;
x ± tv E aX},
(x, v) ED.
Given 9 E L1 (R x r ±, dt d~), consider the following operators of extension: g(±T±(X,v), x ± T±(X, v)v, v),
(E±9) (x, v)
=
(x, v) E D
{ 0, otherwise.
It is easy to check that E± : L1 (R x r ±, dt~) Denote by R± the operator of restriction
-t
£1 (R n x V) are isometric.
Although R± is not a bounded operator on £1(Rn x V) (see [2], [3]), R±Uo(t)f E L1(R x r ±, dt~) is well defined for any f E L1(Rn x V) (see (14.41)). Denote by Xf! the characteristic function of D. We establish the following relationships between Sand $. Theorem 14.2 Assume that X is convex. Then (a) $g = R+Uo(t)SE_g, 9 E L~(R x r _,dt~), (b) Sf = E+$R_Uo(t)f + (1- Xf!)f, f E L~(Rn x V\{O}), (c) $ extends to a bounded operator
~f and only if S extends to a bounded operator on L1(Rn x V).
Remark 14.1 Let us decompose L1(Rn x V) = L1(D)EB£1((Rn x V)\D) . A similar decomposition of course holds for L~(Rn x V\ {O}). Then S leaves invariant both spaces, moreover SIL'((Rn XV)\f!) = Id, so S can be decomposed as a direct sum S = S1 EBI d . Denote R± = R±Uo( . ) : £1 (D) - t
Chapter 14. Inverse scattering and albedo operator
313
Ll(R x r ±, dtdO. We will see in Section 14.4 that R± are isometric and invertible and Rj/ = £± with £± : Ll(R x r ±, dtd!,) - t Ll(O) , £±f := E±flu(!1). Then we can rewrite Theorem 14.2 (a) , (b) in the following way
or even more simply as
with SI = SIL~(!1) as above. Thus $ can be obtained from SI by a conjugation with invertible isometric operators and vice-versa. Remark 14.2 The albedo operator is defined in [lJ in somewhat different manner by (14 .8), provided that u solves (14.7) for t > 0 and u satisfies Ult=o = 0 instead of Ult«o = 0 (in fact, it is assumed that u satisfies a non-zero initial condition, but this can be easily reduced to the case of zero initial condition). The relationship between $ and S established in [1] can be written as R+SUo(t) = $R-Uo(t), which can be obtained as a consequence of Theorem 14.2(b) (or (a)) . Remark 14.3 Some of notations above are a little bit ambiguous. Namely, the expression E+$R_Uo(t)f in Theorem 14 .2(b) seems to depend on t, while the left-hand side of the equality in which it is involved is independent oft. In fact, here Uo(t)f is a function of x, v and t is a parameter, the same applies to R-Uo(t)f . Since the operator $ acts on functions g = g(t, x, v) depending on t as well, we consider now t as a variable and apply $ to the function (t, x, v) f-+ R-Uo(t)f . The result is a function oft, x and v. Next we apply E_ and obtain a function of x and v only. Perhaps a more precise notation in Theorem 14.2(b) would be E+$R_Uo( )f .
An immediate consequence of Theorem 14.2 is that $ determines uniquely k for (J a independent of v and X convex. However, we can prove this for not necessarily convex domains as well independently of Theorem 14.2. (J a,
Theorem 14.3 Let ((Ja , k), (aa, k) be two admissible pairs with (Ja, aa independent of v and denote by X any open bounded set with Cl-smooth boundary with the p;:.operty that (J a, k, aa, k vanish outside X . Then, if the albedo operators $, $ on ax coincide, we have (Ja = a;; , k = k .
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Topics in Neutron Transport Theory
It should be mentioned that in the case where k is of the form k
=
RK + Q with R, Q known and K = K(x, v), the uniqueness of the inverse problem studied in Theorem 14.3 has been investigated in [13] for convex X
under some smallness assumptions that guarantee that the corresponding integral equations can be solved by successful approximations. We would like to mention here also [5], where the problem of determination of k from the stationary albedo operator in the one-dimensional case is considered. The proof of Theorem 14.3 is constructive as well. We study the Schwartz kernel of $ and describe the first two singular terms in it. We show that a a can be recovered from the first term, while the second one determines k, similarly to the proof of Theorem 14.l. Finally, we would like to mention that we have found some analogy between the albedo operator $ and the Dirichlet-to-Neumann map A related to the boundary value problem for the Schrodinger operator or for the conductivity operator \1 . 'Y\1, 'Y = 'Y(x) > 0 [17]. More precisely, denote by A the operator acting on the boundary of a bounded domain mapping the Dirichlet data of the solution to (-Ll + q)v = 0 (respectively \1 . 'Y\1v = 0) to its Neumann data. As proven in [17], A determines uniquely q (respectively 'Y). We found that $ in our case is in some sense an analogue to A or more precisely to the time-dependent Dirichlet-to-Neumann map associated with the wave equation (8;- Llx + q)v = o. It is well-known that there is a close relation between the scattering operator for the Schrodinger equation and A. Theorem 14.2 we proved can be considered as an analogue of this result in transport theory. We would like to mention however, that the Schrodinger equation and the Boltzmann equation have quite different properties. The material in this chapter is taken from [4]. It should be mentioned that the main theorems can be generalized to the case where aa, k depend on t as well.
14.2
The special solution and the scattering operator
An important role in our analysis is played by the following special solutions. Given (x', v') ERn x V\ {O} consider the following problem
8(x - x' - tv)8(v - v'),
(14.10)
Chapter 14. Inverse scattering and albedo operator
315
8 being the Dirac delta function. We will show that (14.10) has unique solution u(t, x, v, x', Vi), with u depending continuously on t with values in V' (R~ x Vv x R~, X Vv' \ {O} ). Moreover, we have the following singular expansion of U. Theorem 14.4 Problem (14 .10) has unique solution where
Uo
= e - Joroo CT,,(x-sv,v)ds uI:( x
- x1 -
_ Jex> e- Jor' CT,,(x-7'v,v)d7' e -
ul =
t) v uI:( V -
u = uo + UI + U2, V')
roo CT,,(X-SV-7'v',v')d7'
Jo
o xk(x - sv, Vi, v)8(x - sv - (t - s)v' - x') ds and
Proof: Pick 'P E
C~(Rn
x V\ {O}) and consider the problem
Wlt«o
=
'P(x - tv, v),
(14.11)
Since min{lvl; (x, v) E 'P for some x} > 0, there exists to = to('P), such that Uo(t)'P = 0 for x E X, t < -to. Then w := U(t
+ to)Uo( -to)'P
(14.12)
solves (14.11) and it is easy to see that w does not depend on the particular choice of to. Applying Duhamel's principle (14.13) we get
Applying Duhamel's formula once more, we obtain
Topics in Neutron Transport Theory
316
where
Wo = U1(t WI
=
W2 =
+ to)UO( -to)
j~to U1 (t -
s )A2U1 (S + to)UO( -to)
jt-to jt U(t- s2)A2U (S2 -sl)A2U (SI + to)UO(-to)
1
S1
For the first term Wo we have
Wo
= e - Jor'+'o
<7
"
(x-sv v)ds ,
_ = (uo(t, x, v,
,.),
(14.14)
where Uo is as stated above and (uo(t,x,v, . , . ),
=
t+to {
Jo
{t+to
Jo
Jv e-
1." (X-TV,V )dTk(x-sv,v',v)wo(t-s,x-sv , v')dv'ds 0 <7"
{rS roo I Jo <7,,(x-Tv,v)dT e - Jo <7,,(X-SV-TV ,v )dT I
Jv e -
x k(x - sv, v', v)
(14.15)
where UI is as stated above. Finally, for W2 we get by changing the order of integration (14.16)
Substituting the formula for
WI,
we obtain
Chapter 14. Inverse scattering and albedo operator
317
JJJ ex>
E(51,X,V",VI)k(x,VI',v)k(x-51V",V',V")
V 0 V
(14.17) where
E( 5, X, V, V
')
= e-
oo is O'a(X-Tv,v)dTe - i O'a(x-sv-Tv',v')dT o . 0
The second integral in (14.17) is in fact over a bounded interval. Since the integrals in (14.17) are absolutely convergent, we can change the order of integration freely. Let us make the following change x' = x - (52 - 5I)V I 51V" in the first integral in (14.17). We get
x k ( x I + ( 52
-
51
) I
v, v,I
X -
x' -
(52 -
5l)VI)
('
')
I
I
cp x, v dx dv d5l.
(
14.18
)
51
Here we suppose that k(x, v', v) is prolonged as 0 for v or v" outside V . There is a singularity above in 51, but the integral (14.18) converge, because we obtained it from the convergent integral (14.17). Denote by M(5I, 52, x, v, x', v')
x k(x'
+ (52 -
51)V I,V' ,
Then we can rewrite (14.18) as
x - x' -
(52 51
5dv'
).
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Topics in Neutron 'lransport Theory
By performing the change x
IvI xk(x '
= x' + (S2 - sdV' + SlV",
we get for Sl
>0
M(Sl,S2,X,V,Xf,Vf)dxdv
+ (S2 -
Sl)V / , v', v") dV'f dv ::;
Ilapllioo,
because 0 < E(s, x, v, v') ::; 1 for s 2: O. Therefore, ME LOO((R+)Sl x RS2
X
Rnx' x Vv'; L1(Rnx x Vv ))
(14.19)
and moreover, for each compact KeRn x V\ {O} there exists to = to(K), such that if (x', v') E K, then M vanishes for Sl > S2 + to and for S2 < -to (provided that Sl > 0). Therefore, the following integral is well-defined
U2 :=
jt
roo U(t _ s2)M(Sl, S2,
-00
Jo
, ,x', v') dS 1 ds 2,
and we have
U2 E C (Rt; L~c(Rn x, x Vv' \ {O}, L1(Rn x x Vv ))) . On the other hand, by (14.16)
W2(t, x, v)
=
r
JRnxV
U2(t, x, v, x', V/)cp(X', v') dx ' dv' .
(14.20)
We are now ready to conclude the proof of Theorem 14.4. We found (see (14.14), (14.15), (14.20)) that the unique solution to (14.11) has the form w = (u(t,x,v, , . ),cp), where u = Uo + U1 + U2 is a distribution with properties as stated above. It is clear now that u solves the transport equation in distributional sense and satisfies the initial condition in (14.10) as well, therefore u solves (14.10). Moreover, this solution is unique because the solution to (14.11) is unique. 0 We will prove next that the wave operators W _, tv+ always exist as operators between suitably chosen spaces. Proposition 14.1 The limits W_, W+ exist as operators between the spaces
W_ : L~(Rn x V\ {O}) ~ L1(Rn x V) W+: L1(Rn x V) ~ Ltoc(Rn x V\ {O}).
Chapter 14. Inverse scattering and albedo operator
319
Proof: Pick fJ(RnxV\{O}) . Sincemin{lvl i (x,v) Ef for some x} > 0, for some to = to(f) we have Uo(-t)f = 0 in X for t > to. Moreover, U(t)Uo(-t)f = U(to)Uo(-to)f for t ;:::: to and therefore W-f = U(to)Uo(-to)f. This proves in particular that the limit W_ : L~(Rn x V \ {O}) -+ L I (Rn x V) (see (14.1» exists as an operator between these two spaces. Next, let us fucg E LI(Rn x V) and a compact set K c Rn x V\{O} and consider [Uo( -t)U(t)g](x , v) for large t and (x, v) E K. We claim that this is independent of t for t > tl with some tl = tl(K). In particular, this would prove that the limit W+ (see (14.5» exists as an operator W+ : LI(Rn x V) --+ Lfoc(Rn x V\ {O}) and W+gIK = Uo( -tl)U(tl)gIK. Indeed, the Duhamel's principle U(t) = Uo(t)
+
lot Uo(t - s)AU(s) ds
(14.21)
implies
Uo(-t)U(t)g = 9 +
lot Uo(-s)AU(s)gds .
(14.22)
Since AU(s)g = 0 for x fJ. X, we have Uo(-s)AU(s)g = (AU( s)g)(x + sv, v) = 0 for (x, v) E K , s > tl = tl(K) . Therefore, Uo( -t)U(t)9IK does not depend on t for t > tl and our claim is proved. <) Now we are a in position to define the scattering operator. Set (14.23) where W+, W_ are as in Proposition 14.1. In fact , as can be seen from the proof of the proposition above, S is well-defined on the wider subset {I i 3to = to(f), such that Uo(t)f = 0 for x E X , t < -to} (the incoming space) . Now it is clear that W+ , W_, S exist in classical sense if and only if these operators given in Proposition 14.1 and (14.13) can be extended as bounded operators on LI(Rn x V). Proposition 14.2 iL(O, x, v , x', v') is the Schwartz kernel of w_ .
Proof: We will prove something more - that u(t, x , v, x', v') is the Schwartz kernel of U (t) W _. Pick cp E C':' (Rn x V\ {O} ). From the proof of Proposition 14.1 it follows that W_cp = U(to)Uo( -to)cp for some large to = to(cp). Therefore, U(t)W_cp = U(t+to)Uo(-to)CP and by (14.12) we deduce that U(t)W_cp = w , where w solves (14.11), i.e. w = (u(t , x , v, . , ),cp). This completes the proof of the proposition. <) Denote by S(x,v,x',v') E V'(Rn x V\{O} x Rn x V\{O}) the Schwartz kernel of the scattering operator S.
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Proposition 14.3 S(x, v, x', v') = lirnt,,
z(t, x
+ tv, v, x', v').
Proof: Let cp E CF(RnxV\{O)). Then U(t)W-cp = (ii(t,x,v, ., ),cp) and Uo(-t)U(t) W-cp = (ii(t, x +tv, v, . , . ), cp). Therefore, for any compact K c R n x V\{O) we have ScpIK= limt+oo(fi(t,x tv,v, - ,- ) , p ) ( ~Note . that, according to the proof of Proposition 14.1, in the last limit it suffices to take t > to(K), so the limit exists trivially. 0
+
Proposition 14.4
S(x - x')S(v - v') +
J +e- ~STrnu.(x+~v,v)dr ( A Z ~ ) ( Sx ,+ SV, V, x', v')ds. -M
Proof: It follows from (14.13) that
Pick up f E L:(Rn x V\{O)) and fix a compact K there exists to = to(f, K), such that
C
R n x V\{O). Then
s f l=~~ O ( - ~ O ) ~ ( ~ ~ O ) ~ O ( - ~ O ) ~ J K
where w solves (14.11) with p = f .
0
14.3 Reconstruction of a,, k from S Assume that we are given the scattering operator S corresponding to an admissible pair (u,, k). We will show in this section how one can recover na, k constructively. In particular, this will prove Theorem 14.1.
Chapter 14. Inverse scattering and albedo operator
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We will show next that the singular expansion of the special solution
u(t, x, V, x', v') established in Theorem 14.4 implies a similar expansion of the scattering kernel S. Theorem 14.5 We have S = So + Sl + S2, where the Schwartz kernels Sj(x, v, x' , v') of the operators Sj , j =-0,1,2 satisfy
S o = e- J+oo O"a(X-Tv,v)dT u"( x - x ') u"(V -00
S1x k(x
/
+00 - f+oo e ·
O"a(X+Tv,v)dT -
e
-
V
')
J.+oo O"a(X+SV-TV ,,V' )dT 0
-00
+ sv , v' , v)8(x -
x'
+ s(v -
v')) ds
and Proof: The proof follows by substituting u = Uo + U1 + U2 from Theorem 14.4 into the limit in Proposition 14.3 or the integral representation of S found in Proposition 14.1. As already mentioned above, for (x , v , x' , v') E U cc Rn x V\ {O} x Rn x V\ {O} the limit in Proposition 14.3 trivially exists and the integrals in Proposition 14.4 are taken over bounded intervals. (; We are ready now to complete the proof of Theorem 14.1. The idea of the proof is the following. Suppose we are given the scattering operator S corresponding to an unknown admissible pair (aa , k) . Then we know the kernel S = So + Sl + S2. It follows from Theorems 14.4 and 14.5 that So is a singular distribution supported on the hyperplane x = x' , v = v' of dimension 2n, Sl is supported on a (3n+1)-dimensional surface (for v i= v'), while S2 is a function. Therefore, Sj , j = 0 , 1, 2 have different degrees of singularities and given S = SO+Sl +S2 , one can always recover So and Sl . From So one can recover the X-ray transform of aa and therefore aa itself, provided that aa is independent of v. Next, suppose for simplicity that aa, k are continuous. Then for fixed x, v , v' with v i= v' , Sl is a delta-function supported on the line x' = x + s(v - v') , s E R with density a multiple of k(x + sv , v', v). Therefore, one can recover that density for each s and in particular setting s = 0 we get k( x , v', v) . Pick a function cp E C.;"'(Rn) with cp(O) = 1, cp(x) = 0 for Ixl > 1 and o :S cp :S 1. Fix a compact set KeRn x V\ {O} and let X E C.;"'(Rn x V\ {O}) be such that X = 1 on K and 0 :S X :S 1. For 10 > 0 sufficiently small set
x' ) (v -
VI) X(x,v). X ¢g(X ,V, XI,V')=cp ( -10cp -10-
(14.24)
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Note that 4, = 0 for ( X I , v t ) outside some other compact subset K t of Rn x V\{O) for E sufficiently small. Proposition 14.5 With 4, as above we have lim
E+O
J J ~ ( xv ,,
00
XI,v
~ ) ~ v( , XxI ,,v l )dxldvl = e-
J-
a,(x-TW,V)~T x ( x ,v )
in L 1 ( R n x V), whew the integral is to be considered i n distribution sense. Proof: Note that a priori the formal integral above is a distribution in D'(Rn x V\{O)), but we will show that by Theorem 14.5 in fact it belongs to L 1 ( R n x V) and the limit above holds in the same space. For So we have
Next,
5
//IJ
X ( x ,v )
(G) + k(x
sv, ul,v ) ds dvl dx dv
We have used above the fact that the integral in s is taken over some bounded interval [-A,A] with A > 0 depending on x and X, and we denoted W = { v ; 3x, such that ( x ,v ) E suppx). The last integral is taken over the bounded set
DE= { ( x , v l ,v ) , x E X ,v E W , Iv - v'l
< E).
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323
Let us estimate the measure meas(D,). We have meas(D,) =
J
W
11 X
dv' dx du = Cen (u-v'(<E
JW
dx dv = C'E".
Therefore, in (14.26) we have a locally integrable function (see (ii), (iii) in Section 14.1) and the integration is performed over a set D, with meas(D,) + 0. Since the Lebesgue integral is absolutely continuous with respect to the Lebesgue measure, we get that
in L1 (Rnx V). Finally, we have
JSIJJ 5
S2(x, v, x', vl)&(x, v, XI, vl) dx' dv' dx dv
lS2(x,v,x',u')l dxdvdx'dv' Ec
with E, = {(x, v, x', v'); (x, v) E supp X , lx - x'l 5 E, Iv - v'l 5 E). There exists EO > 0 such that for 0 < E < EO we have E, c E,, C Rn x V\{O) x R n x V\{O) and S2is an integrable function on E,, by Theorem 14.5. As before, it is easy to see that meas(E,) = O(E'~)+ 0 as E + 0. Therefore, (14.28) tends to 0 and we obtain
in L1(Rn x V). Now, (14.25), (14.27) and (14.29) together complete the proof of Proposition 14.5. 0 Assume now that a, is independent of v. We deduce from Proposition 14.5 that one can recover
for a.e. (x, v) E Rn x V. Since V is an open set, we see that we know (14.30) for a.e. (x, v) E Rn x U,where U is a small neighborhood of some vo E V\{O). Thus we can recover the X-ray transform
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of l7a (x) for a.e. (x, w) E Rn x U, where U is a small neighborhood of vo/lvol in sn-l The latter is sufficient to recover l7a (see e.g. [9]) . We note that in the particular case where for any w E sn-l, the velocity space V contains some v of the kind v = rw with r = r(w) > 0, we can recover the X-ray transform (14.31) for a.e. (x,w) ERn x sn-l and therefore we can write an explicit formula [9] for l7a (x). We proceed next with the reconstruction of k(x, Vi, v). Choose two functions cP E C.;"'(Rn) with ~ cP ~ 1, cp(o) = 1, cp(x) = for Ixl > 1 and CPl E C.;"'(R) with CPl(s)ds = 1, ~ CPl· Set W = {(v',V) E V x Vi v =J. 0, Vi =J. 0, v =J. Vi} . For 101> 0, 102> set 'l/Jc l,c2 equal to
J
xcP ( -1 ( X 102
-
X
I
-
°
°
°°
(x-xl).(V-V Iv - v'12
I)(
V -
V
I)))
.
(14.32)
Proposition 14.6 With 'l/Jc l,c2 as above we have
<1a(x+rv,v)dr - .fc+oo <1 a(x-rv',v')dr ( = e- .fc+oo 0 e 0 k x,v ,v I
)
where the integral is to be considered in distribution sense and the limit holds in Ltoc(Rn xW). Proof: First, note that
J
So(x, v, x', V ' )'l/Jcl,c2 (x, v , x', Vi) dx'
because Solv,ev'
=
= 0.
.I
=
°
in xW,
(14.33)
Next, for Sl we get
E(s, x, v', v)k(x
+ sv, v', v) cll CPl( -
:1) ds.
) = e - Joo He r e E- (s, x, v',v • <1a(x+rv,v)dr e - JofOO <1a(x+sv-rv',v')dr . It·IS easy t 0 see that the mapping s --+ E(s, x, v', v)k(x + sv, v', v) E Ltoc(Rn x V x V)
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325
is continuous. Therefore, lim lim I SI (x, V, X',
€'1 --+0 £2--+0
V')'ljJC1.C2 (x,
V, X', v') dx'
lim IE(s, x , v', v)k(x + sv, v', v)!
Cl->O
10 I
'PI(-~) ds ci
E(O, x, v', v)k(x, v', v).
(14.34)
Finally, choose a compact set K eRn x Wand consider
I IS2(X, v, x', V')'ljJcl.C2 (x, v, x', v') dx'i dxdv'dv K
<
lI'PIIL:x> III S2(X,V,X',v')1 Ci _
(14.35)
KX'
X'P(~2 (x-x'- (x-I:'~'~~2-v')(V-v')))dX' dxdv'dv. Note that on supp 'ljJ c l.C2 we have Ix - x'i ~ C max {Cl, cd, so for Ci and 102 bounded, x' belongs to some compact set X' The integration in (14.35) is taken over the set
"
(x,x,v,v)EFc2
:=
(') ' (x-x') , (v-v')( v-v ')1 XxKn {I x-xIv-v'1 2
For fixed x, v', v the last set is a cylinder with radius that meas(Fc2 ) = O(c2'-I) and therefore
102,
<102
}
.
thus we conclude
I IS2(X, v, x', V')'ljJcl.C2 (x , v, x', v') dx'i dxdv'dv K
<
II'PIII
I IS2(X, v, x', v')ldx'dxdv'dv
----7
0,
(14.36)
Ci F£2
because the Lebesgue integral is absolutely continuous with respect to the Lebesgue measure. Combining (14.33), (14.34) and (14.36) we complete the proof of the proposition. <> Assume now that we are given the scattering operator corresponding to an admissible pair (lT a, k) with lTa = lTa(x). By Proposition 14.5, one can recover lTa(x) . Next, by Proposition 14.6 we can explicitly recover
e _ Joroo Ga(x+Tv)dT e _ Joroo Ga(x-TV')dTk( x, v , , v )
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almost everywhere in Rn x V x V. Since lTa(x) is already known, we get k(X,V',V) for a.e. (X,V',V). Finally, we would like to mention that Propositions 14.5, 14.6 can be written in terms of the operator S itself rather than in terms of its distribution kernel. Let ¢g be the same as in (14.24) and consider the function ¢g(y, w , x, v), where y, ware regarded as parameters (i.e. in (14.24) we replace (x, v, x', v') by (y, w, x, v)). Then Proposition 14.5 is equivalent to A. ( lim S 'f'g y,w"
g-+O
)1 y=x,w=v = e- foo ua{x-rv,v)dr X ( X,V ) 00
in LI(Rn x V) . Similarly, Proposition 14.6 can be rewritten as
=
I -
ua{x+rv,v)dr - J.oo ua{x-rv',v')drk( e J.oo 0 e 0 x,v I ,v ) p( v') dvI
.
V
in Lfoc(Rn x V \ {{O} U suppp}) for any p = p(V')~(V \ {o}).
14.4
The albedo operator
Assume that X is convex and ax is CI-smooth. Consider the functions T±(X, v) and the operators E±, R± defined in the Introduction. It should be noted that T± have the properties T±(X + tv, v) = T±(X, v) =f t and (x ± T±(X, v)v, v) E r ± for any (x, v). Using this property, we can show that E± are closely connected to the solution of the following boundary value problem
(at - To)v vlRxr±
o in R 9
x X x V, (14.37)
Indeed, taking into account that (14.38) we see that the solution to (14.37) is given by v = Uo(t)E±glxxv. We consider in (14.37) To as a differential operator in X x V. As mentioned in the Introduction, E± : LI (R x r ±, dt df.) -+ LI (Rn x V) is isometric, i.e. (14.39)
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327
Equality (14.39) follows easily by making a change of variables in the corresponding integral. Indeed,
Let us choose new variables t = fT+(x, u), y = x fT* (x, u). Then (y,u) E I?* and dx = dtlu . n(y)ldp(y), thus we get
which proves (14.39). Denote by xn the characteristic function of R (see (14.9)). Then the following property holds
(see also Remark 14.3). Taking into account (14.37), we conclude that
Note that R* are unbounded operators on L1(Rn x V). We refer to [2], [3] for more precise results and trace theorems. We are not going to make use of these trace theorems however (except in the proof of Proposition 14.9), because we will always apply I& to time dependent functions like Uo(t)f or U(t) f and will consider the result as a function of both variables x and t belonging (locally) to L1( R x I?*, dt &). Then R*Uo(t) f is well-defined according to (14.41) and for R*U(t) f we have: Lemma 14.1 &U( ) : L1(Rn x V) --t Lt,,(R; L1(I?*, &)) is continuous. More precisely, for each a > 0 we have
+
Proof: Given f E L1(Rn x V), set f = fl f2 with fl = Xnf, f2 = (1 -xn) f . Using Duhamel's principle (14.13), we see that U(t) f2 = Uo(t)f2 and thus by (14.41), &U(t) f2 = 0. For fl we have by using (14.21) and
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328 (14.41)
f:a IIR±U(t)fI IILl(r±.de)dt
:S: IlfIIILl(RnxV) + fa 11ft R±Uo(t - s)AU(s)fI dsll -a
0
dt £l(r±.~)
:S: IlfIIILl(RnxV) + f:a f:a IIR±Uo(t -
s)AU(s)fIIILl(r±.~)dsdt
IlfIIILl(RnxV) + f:aJ:a IIR±Uo(t -
s)AU(s)fIIILl(r±.~)dtds
=
:S: IlfIIILl(RnxV) + J:a IIUo( -s)AU(s)fdILl(Rnxv)dS :S: (1 + 2allo-pllLooeallupllLOO) IlfIIILl(RnxV)' <> Lemma 14.2 R-U(t)fIR+xr _
=
R-Uo(t)fl~xr _
for any f E Ll(Rn x V).
Proof' We have to show that
By inspecting the proof of Lemma 14.1 we see that it suffices to prove that
which is equivalent to
In order to complete the proof, it is enough to observe, that
R_Uo(t)hI R+xr _ = 0 for any h with h(x, v) = 0 for x ¢. X. <> Given g E L~(RiLl(r_,d~)), consider the problem (14.7) Proposition 14.7 Problem (14.7) has a unique solution in C(RiLl(X x V)) given by u = U(t)W_E_glxxv,
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329
Proof: Note first that the uniqueness follows from the fact that the homogeneous problem (with g = 0) has only a trivial solution, because the transport operator with boundary conditions ulRxr- = 0 generates a continuous semigroup of solution operators. Next, note that if to is such that g = 0 for t < -to, we have Uo(t)E-g = 0 in X x V for t < -to and moreover, U(t)Uo(-t) E-g = U(to)Uo(-to)E-g for t > to, so although E-g does not necessarily belong to L,!(Rn x V\{O)) (see (14.1) and Proposition 14.1), the limit W-E-g trivially exists. Set w = U(t)W-E-g = U(t + to)Uo(-to)E-g. Then w clearly solves the Boltzmann equation in R x X x V. We have that R-wlt,-to = 0 and by Lemma 14.2 and (14.38), R-wit>-to = R-Uo(t)E-glt>-t, = glt>-to = g. Therefore, w satisfies the boundary condition as well. Thus setting u = wlxxv, we get a solution to (14.37). 0 We see now that the definition of $, given in (14.8) $g = R+U, $ : L:(R; ~ l ( r - ,cy)) +L;,,(R;
~ l ( r +cy)), ,
(14.42)
u being the solution to (14.7), is correct. Indeed, by Proposition 14.7, $g = R+U(t) W-E-g and by Lemma 14.1, $g E L;,,(R; L1(I'+, @)). We note that in fact, $g is well-defined also for g E L1(R x I?-, d t e ) with g = 0 for t << 0, and then $9 E L1((a, oo) x I?+, dt @) for any a E R , but as Theorem 14.2(c) shows (see the proof below), $ extends as an operator $ : L1(R x I?-, d t e ) -L1(R i x I'+,dt @) if and only if the scattering operator exists a s a bounded operator on L1(Rn x V). Proofof Theorem 14.2: Consider first (a). Pick g E LE(R x r-, dt @). Then E-g E Li(Rn x V\{O)) and SE-g is well defined. By (14.42) and Proposition 14.7 we have $g = R+U(t)W-E-g. Denote
We claim that
for any compact K
c Rn x V\{O). Indeed,
uo(t)SE-g = lirn Uo(t - s)U(s)W-E-g s-m
and the limit exists in LL(Rn x V\{O)). We have (see (14.22))
(14.44)
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330
with Uo(t-r)AU(r)W_E_g = (AU(r)W_E_g)(x-(t-r)v, v). For (x, v) E n+ this function vanishes provided that t - r < O. So in fact the integral in (14.45) is taken over the interval 0 ~ r ~ t only and therefore (14.45) is independent of s for s ~ t. Therefore, we can put s = t in (14.45) in order to get the limit (14.44) which implies immediately (14.43). Let us now apply R+ to both sides of (14.43) to get R+Uo(t)8E_g
= R+U(t)W_E_g = $g .
Consider (b). Pick f;(Rn x V\{O}) and set 9 = R_Uo(t)f. Then we have 9 E L~(R; L1(r _, d~)) and E_g = xnf (which is true whenever E_g E L~(Rn x V\ {O}). An application of (a) yields $R-Uo(t)f = R+Uo(t)8Xnf by (14.40) . Applying E+ to both sides and using again (14.40), we get (14.46) Now, since for the special solution ii we have ii = c5(x - x' - tv)c5(v - v') in (Rn x V\ {O}) \ n, we get 8(1 - xn)f = (1 - xn)f for any f. On the other hand, by Proposition 14.4, (1 - xn)8f = (1 - xn)f for any f. In other words, 8 leaves L~(n), L~((Rn x V\ {O}) \ n) invariant and xn8xnf = 8 f - (1- xn)f. Substituting this into (14.46), we complete the proof of (b). Finally, (c) is an immediate consequence of (a), (b), (14.39) and (14.41). <>
14.5
Reconstruction of O"a, k from $. The nonconvex case
In this section we prove Theorem 14.3. In the case where X is convex, the uniqueness result in Theorem 14.3 is an immediate consequence of Theorems 14.1 and 14.2. If X is not convex, then one can still deduce Theorem 14.3 from the previous two theorems using an argument from [18], where the Dirichlet-to-Neumann map is considered (see Proposition 14.9 below) . Namely, one can show that $ = :£ entails ii = ii outside X x V and therefore, by Proposition 14.3 we could conclude that (Ta = a;;, k = k. We will give however another proof of Theorem 14.3 as well, that implies a constructive procedure for recovering (Ta, k and describes the Schwartz kernel of the albedo operator $. We assume in this section that X is an open bounded set with C 1 _ smooth boundary, not necessarily convex. First we will show that (14.7) still have unique solution in this more general situation. In what follows we need the semigroups Uo(t), U1(t), U(t) (see e.g. [20], [22]), related to the
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331
solution of the following problem
o in R
x X x V,
o Ult=o
f
=
(14.47)
Ti being To, Tl and T, respectively (regarded as differential operators). More precisely, To, Tl and T, acting on functions vanishing on r _, extend to generators of strongly continuous semigroups Uo (t), Ul (t), U (t) on L 1 (X X V) and the solution to (14.47) is given by u = Ui(t)f. It is easy to check that we have the following explicit formulae
Uo(t)f U1(t)f
f(x - tv, v)8(x, x - tv)
=
e-
fa'u
a (x-sv,v)dS
(14.48)
f (x_tv,v)8(x,x_tv)
(14.49)
where if px + (1 - p)y E X for each p E [0,1],
I,
8(x,y) =
{ 0,
otherwise.
Let us modify a little bit the definition of T± given in the Introduction. Set
T±(X,v) = min{t
~
0; x±tv E aX},
(x,v) x V\{O} .
If X is convex, then the definition given above agrees with that proposed in the Introduction. Using L , we can write explicitly the solution of (14.7) in the case where T = To or T = T1. For the case T = Tl we have that the solution of (14.7) reads u = G_(t)g, where
G±(t)g := e
±
(±
fT±(X .")
Jo
era
X
)d SV,V
S
get ± T±(X, v), X ± T±(X, v)v, v). (14.50)
For X convex and IJa = 0, we have G±(t)g = Uo(t)E±g. It is not hard to see that the following generalization of (14.39) holds (14.51) and moreover, G_(t) : Ll(R x r _, dtdf,) ----; Ll(X x V) is strongly continuous in t. Similar statements hold for G + (t) as well if we restrict our considerations outside a small neighborhood of v = 0, because the exponential in (14.50) may not bounded in this case as v ----; O. We have the following generalization of Proposition 14.7 to the case where X is not necessarily convex.
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Proposition 14.8 Given 9 E L~(R; LI(r _, d~)), problem (14.7) has a unique solution u E C(R;LI(X X V) given by
u
= G_(t)g +
[too D(t - S)A2G_(s)gds.
(14.52)
Proof: First, observe that the integral above is taken over a finite interval = 0 for t < -to. It is easy to see that u, given by the formula above, satisfies the Boltzmann equation in X x V in distribution sense and belongs to C(R; LI(X x V)). For t < -to we have that G_(t)g and the integral above vanishes, thus ult<-to = O. Finally, UIRxr _ = G-(t)gIRxr _ = g, because U(t) satisfies homogene~s botmdary conditions on R x r _. Notice that the requirement 9 E L~(R; U(r _, d~)) can be relaxed to 9 = 0 for t «0. <> Following the proof of Lemma 14.1 and using (14.48), we can prove that R+U(· ) : LI(X X V) --t Ltoc(R; LI(r +, ~)) is continuous. Thus, using this fact and Proposition 14.8 we can define the albedo operator in this case as well by (14.8). Next we prove Theorem 14.3 by showing that $ determines uniquely the special solution u outside X x V. Although the reconstruction procedure described after this proposition implies Theorem 14.3 as well, we include Proposition 14.9 because it suggests much shorter way of proving Theorem 14.3.
[-to, tJ, where to is such that 9
Proposition 14.9 $ determines uniquely the special solution side X.
u for x
out-
Proof: Here we follow essentially [18], where the Dirichlet-to-Neumann map related to a second order elliptic equation is considered. Let (aa, k), (aa, k) be two admissible pairs supported (with respect to x) in X and denote by T, T, u, ii, etc. the operators T, the special solutions U, etc., related to (a a, k) and (a a,k), respectively. Choose cp:;'" (Rn x V \ {O}) and set w:= (UCt,x,v, " . ),cp), w:= (u(t,x,v, ,. ),cp), i.e. w, w solve (14.11) with T = T, T = T, respectively. Since cp E D(T) = D(T), we have w E D(T), w E D(T) for each t (see (14.12)), therefore (see [2], [3]) the traces wlr±, wlro±- are well defined as elements in LI(r±,~) depending continuously on t. Let v solve
(at - T)v
0 in R x X x V
(14.53)
333
Chapter 14. Inverse scattering and albedo operator Set xEX
V,
u=: {
w,
x
tJ.
(14.54)
x.
Asswne that $ = $. Since ,w clearly solves the problem and $ = ii, we get from (14.53), (14.55), that VIR xr + = WIRxr +' therefore
(at -
o inRxXxV
T)w
WIRxr_
WIRxr_
O.
Wlt«o
(14.55)
VIRxr ± = WIRxr ±.
(14.56) Combining (14.54) and (14.56) we deduce that u, which is absolutely continuous function along the rays s f-+ (x + sv, v) with possible jwnps on r _ u r +, in fact has no jwnps on these rays. Since both v and w solve the Boltzmann equation (14.1) in X x V and outside X x V, and there is no jwnp at the boundary, we conclude that u satisfies (14.1) everywhere. Therefore, u = w, because the solution to (14.11) is unique. In particular (see (14.54)) we get w = wfor x tJ. X. 0 Theorem 14.3 is now an immediate consequence of Propositions 14.9 and 14.3. Note that the proof of Theorem 14.3 provided above is not constructive. Below we will give explicit formulae for the reconstruction of (Ta, k which in particular provides another proof of Theorem 14.3. This proof is based on an analysis of the Schwartz kernel of the operator $. A priori this kernel is a distribution in 'V'(R x r + x R x r _) . Denote by b1 the Dirac delta function on R 1 and by by (x) the Delta function on a X defined by (by, t.p) = t.p(y) . Theorem 14.6 The Schwartz kernel of$ has the form a(t-t/ , x, v, x', v'), i.e. formally ($g)(t, x, v) = JRxr _ aCt _t/, x, v, x', V')g(t', x', v')dt' dJl(xl)dv ' with a = ao + a1 + a2, where aj(7, x, v, x', Vi) ((x, v) E r +, (x', Vi) E r_) satisfy
a1
=
J
e
-
1.
8
O'a(x-pv,v)dp -
0
e
1.
, T _ ( X - 8 V,V)
(
,
')
O'a x-sv-pv ,v dp
0
X b1 (7 - S - L(x, -SV, v'))k(x - sv, V', V)b{X-SV-T_(X-SV,v/)v'}(X ' ) x 8(x - sv,x)ds.
Topics in Neutron 'nansport Theory
334
and
In(x') . V'I-la2 E LOO(r _; Lfoc(R,.; Ll(r +, d~))) . Proof: The proof is similar to that of Theorem 14.4. Fix 9 E C~(R x
r _)
and let U solve (14.7). Combining (14.52) with Duhamel's formula, we get U = Uo + Ul + U2 with
Uo = G_(t)g, Ul = fooo Ul (s)A 2G_(t - s)gds oo U2 = f~oo fo U(t - s2)A 2Ul (Sl)A 2G_(S2 - st}gds l dS2. By (14.50),
R+uo =
r
lRxr_
ao(t - t', x, v, x', v')g(t', x', v') dt' dJ.L(x') dv',
where the integral is to be considered in distribution sense. For Ul we have by (14.48), (14.50),
oo e - Jor" C7 (x-pv ,v)dp e - Jor T a
JJ V
(",-"V , vI)
I
C7 a
l
(x-sv-pv,v )dp
0
xB(x - sv, x)k(x - sv, v', v) (14.57)
xg(t - s - r-(x - sv,v'),x - sv - r-(x - sv,v')v',v')dsdv', thus
R+Ul =
r
lRxr_
al(t - t', x, v, x', v')g(t', x', v') dt' dJ.L(x') dv'.
Next,
U2 =
J~DO U(t -
s2)A2Ul(S2, " .) ds2·
(14.58)
Using (14.57), we get
(A2Ul)(S2, x, v)
J
= v
J~ Jv E(sl> x,v", v')k(x, v", v)k(x -
slv",v',v")B(x,x - SlV")
Xgs2-sl-7(X-SlV " ,v') ,X-Slv" - 7( _ x-slv" ,v' )V,V ( ")
Chapter 14. Inverse scattering and albedo operator
335
with fa
- - J,0 E( s,x,v,v') -e Set y' =
fT
(x-sv,v')
I
,
J, C1 a (X-SV-pv ,v )dp e o .
C1 a (x-pv,v)dp -
SlV". Then
X -
(A 2U l)(S2,X,V) _
-
00
II
o . v· x
/
xB(x, y/)g(S2 -
-n
Sl
E(Sl,X,
Sl -
s,
x-y'
I
,v)k(x,
L(Y', Vi), y' -
s,
x-y'
T _(y',
I
I
,v)k(y,v,
s, )
x-y'
V')V ' , Vi) dy'dv' ds l .
Let us make the change y' f--t (x', Xl), where x' = y' - L(Y', V')V ' E ax, Xl = T _ (y/, Vi). This change is smooth except on a closed set of measure zero corresponding to y' such that the ray {y' - pv' , P E (O,L(Y/,V ' ))} is tangent to ax at some point. One can first integrate outside a neighborhood of the singular set with measure c > 0, where we have dy' In(y/) . vlldf.L(x/)dxl, and then let c ---; O. Thus we get (A 2U 1)
xk(x
I
+ XIV,I v,I
x' -
X -
Xl Vi
)g(S2 -
Sl - Xl,
I
I
I
x, V ) ~ dXl ds 1 . (14.59)
Sl
Here ~/:= In(x / )· vlldf.L(x/)dv' . Denote by M(SI,82,X,V,X I ,V' )
sln E( Sl, X, v", v')k(x, v", v)k(x' where we have set v"
rr
= (x - x' M(sl, 82,
iviRn
+ Sl Vi, Vi, v")B(x, x' + (82 - Sl)V / ), (82 -
Sl)V')/Sl. It is easy to see that
x, v, x', v')dx dv ::;
Iia-plll"" .
By (14.58) and (14.59) we have U2
=
.It-00 /00 / r_ / U(t -
s2)M(sl, S2 -
t/, ,
,x', Vi)
0
xg(t/, x', v')dt' ~' dS l dS2
=
/L /
a2(t-t/,x,v,xl,vl)g(t/,xl,vl)dtldf.L(x/)dv'
(14.60)
Topics in Neutron Transport Theory
336 with ( T,X,V,X,v I ') a2
JJ T
In(x') . v'I
00
U(T - 52)M(sl, 52, . , . , x', v' )ds 1d52 . (14.61)
-00
0
By (14.60),
In(x ' ). v/I-1a2 E C (R,.j LOO(r _j Ll(Xx x Vv))) .
(14.62)
By (14.62) and the remark after the proof of Proposition 14.8 we obtain
/ In(x') . v I- 1
1:
IIR+a2(T, ,., x', v' )II£1(r +,~)dT ::; C(a)
for any a> 0 and all (x', v') E r _. Setting a(T, . , ., x', v') = R+a(T, ., . , x' , v') we complete the proof of the theorem. <> We proceed with explicit formulae relating the kernel of $ and (Ta, k. Next two propositions are analogues of Proposition 14.5 and Proposition 14.6. Recall again that for each v the set of x, for which L(X, v) has a jump, is of measure zero. Thus the function T _ is smooth on r + outside of a closed set of measure zero. Choose 0 ::; X E c;:, (r +) supported outside that singular set. Let 0 sufficiently small set
l )
lim
10.-0
JJ
a( T, x, v, x', v')¢g( T, x, v, x', v') dT dp,(X/)dv '
r+ =e
-
fr
(X,tJ )
T
0
-
<7 a
(x-sv,v)ds (
XX,V
)
(14.63)
in Ll (r +, d~) , where the integral is to be considered in distribution sense. Proof; We have a = ao + al + a2 with aj as in Theorem 14.6. It is clear that ao satisfies (14.63). For al we have (compare with (14.26))
o
<
r r al(T,x,v,xl , v/)¢g(T, X,v,xl,vl)dTdp,(x/)dv'
ir+iL
Chapter 14. Inverse scattering and albedo operator
XCPl (S
337
+ L(X - SV, v') - L(X, v))k(x - sv , v', v)x(x, v) ds dv' dE,
<
C i k + ! X(X'V)cp(V~V')k(X-SV'V"V)dSln(X).vldfL(X)dVdV'
<
C' ! w i i cp(v~v') k(x,v',v)dxdv'dv
->
0,
as c
->
(14.64)
0,
by the same arguments as in the proof of Proposition 14.5. Finally,
! !! !! ! v'I- Ia2(
adT, x, v,x', V')¢",(T, X, V, x', v') dTdfL(X')dv' dE,
r+ -r_
<
!In(x') , v'I- 1 Ia 2 (T,X ,V,X',v')I¢,,,(T,X,V,X',V')d7dE,'dE,
r+r _
<
1
In(x') .
7,
x, v, X' , v')1 dT dE,' dE,
E. ->
0,
as c
->
0,
(14.65)
where E", = {(T,X, v, x', v') E R x r + x r _; ITI ~ A, (x, v) E sUPPX, Ivv'l ~ c} with some A = A(X, CPl) > O. Clearly, meas(E",) -> 0, as c -> 0, where meas(E",) is associated with dTdE,' dE,. On the other hand, by Theorem 14.6 the integrand in the last integral is a £I-function. AB before, we conclude from this that the limit in (14.65) is zero, as stated. Combining (14.64), (14.65) we complete the proof. <> By Proposition 14.10 one can recover the X-ray transform of CTa(X), provided that CTa is independent of v and therefore one can recover CTa itself. We proceed with the recovery of k. Next proposition is an analogue of Proposition 14.6. Let '1/J"'l' ' 2 and W be the same as in Proposition 14.6.
338
Topics in Neutron Transport Theory
Proposition 14.11 We have
lim lim G+(O)!! a(t - t/, X, v, x', v') Rax
0' -->0 02-->0
=e -
fc
/ T- ( Z , V
0
)
(
1
')
O"a X-PV ,v dP
') k ( x,v,v
(14.66)
where the integral is to be considered in distribution sense and the limit holds in LfoAX). Remark 14.4 The restriction of'ljJo, ,02 (x-tv , v , x'_t'v / , v') on R t x r + x Rt' x r _ \ {v = v'} is not necessarily a function of compact support on that variety, but as will be seen from the proof of Proposition 14.11, the formal integral above is well defined. Operator G+(O) above (see (14 .50)) is applied to the formal integral considered as a function oft, x, v. Proof: Note first that for v =I- v' we have ao = O. Next, for al we get
!! ! E(s,x,v/,v)k(x-sV,V/,V)B(x-sv, X)ol,CPl(t~S)ds,
al (t - t/, x, v, x', v') 'ljJo,,02 (x - tv, v, x' - t'v / , v/)dp,(X/)dt'
R
ax
=
f' ( x-sv ,v' ) ) = e - J,f80 O"a(x-pv ,v)dp e- o J, O"a (x- s v-pv ,v )dp Fun were h . cE( s, x, v',v tion s --+ E(s,x , v/,v)k(x - sv,v/,v)B(x - sv,x) is integrable with values in Ll(r+ x VV/ d1, dv/). Therefore, as Cl --+ 0, the limit above exists in Ll (Rs x r + x VV/, ds d1, dv / ) and we have I
I
lim lim !!al(t-t/,x,v,x/,v / ) Rax
0' -->002-->0
E(t , x, v', v)k(x - tv, v', v)B(x - tv, x).
(14.67)
By applying G+(O) to both sides of the equality above we get that (14.66) holds with a = al'
Chapter 14. Inverse scattering and albedo operator
339
Finally, let us fix a compact set K c r + x Vv ' that does not intersect the varieties v = v', v' = O. Then for any a > 0 we have a
JJ JJ
l l a2(t-t ,x,v,x ,v' )
-a K
R
ax
JJIn(xl)·v/l-lla2(t-tl,x,v,xl,vl)ldt/~~/dt, a
<
(14.68)
-aF£2
with FC2 a set of measure tending to 0, as C2 -+ 0 (compare with (14.36)). By performing the change T = t - t' and using Theorem 14.6, we see that (14.68) tends to 0, as c2 -+ O. Therefore, (14.66) with a = a2 converges to o in LtocCR+; LI(K)). A straightforward generalization of (14.51) implies that G+(O) : Ltoc(R x r +, dt~) -+ Ltoc(X x V) is continuous. Thus, applying G+(O) we get from (14.68) limc2--+o G+(O)
JJ R
{
ax
aCt - t', x, v, x', v') (14.69)
x'lj;cl.c2(X - tv, v, x' - t'v ' , v') d/-L(x ' ) dt' = 0
in Lt"c(X). Combining (14.67) and (14.69), we complete the proof of Proposition 14.11. 0 Now the reconstruction of U a and k goes along the following lines. Given the albedo operator $, we first recover ua(x) (provided that U a depends on x only) by using Proposition 14.10. Next, since U a is already known, we know G+(O) and the exponential factor in (14.66), so by Proposition 14.11 we can recover k( x, v' , v) almost everywhere in X x V xV. Finally, we note that one can rewrite Propositions 14.10, 14.11 in terms of the operator $ rather than in terms of its distribution kernel in a manner similar to that at the end of Section 14.3.
References [1] P. ARIANFAR AND H. EMAMIRAD. Relation between scattering and albedo operators in linear transport theory. Transp. Theory Stat. Phys. 23(4)(1994) 517-531.
340
Topics in Neutron Transport Theory
[2J M. CESSENAT. Theoremes de trace £P pour des espaces de fonctions de la neutronique, C. R . Acad. Sci. Serie 1. 299(1984) 831-834. [3J M . CESSENAT. Theoremes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Serie 1. 300(1985) 89-92. [4J M. CHOULLI AND P. STEFANOV. Inverse scattering and inverse bound-
ary value problems for the linear Boltzmann equation. Comm. Part. Diff Eq. 21(5&6)(1996) 763-785. [5J K. DREssLER. Inverse problems in linear transport theory. Eur J. Mech. B Fluids. 8(4)(1989) 351-372. [6J H . EMAMIRAD. On the Lax and Phillips scattering theory for transport equation. J. Funct. Anal. 62(1985) 276-303. [7J H. EMAMIRAD AND V.PROTOPOPESCU. Relationship between the
albedo and scattering operators for the Boltzmann equation with semi-transparent boundary conditions. Math. Methods. Math. Sci., to appear. [8J J. HEJTMANEK. Scattering theory of the linear Boltzmann operator. Comm. Math. Phys. 43(1975) 109-120. [9J HELGASON. The Radon Transform. Birkhiiuser, Boston, Basel, 1980.
[lOJ M. MOKHTAR-KHARROUBI. Limiting absorption principle and wave operators on £1(f-L) spaces. Applications to transport theory. Funct. Anal. 115(1993) 119-145.
J.
[l1J V. PROTOPOPESCU. On the scattering matrix for the linear Boltzmann equation. Rev. Roum. Phys. 21(1976) 991-994. [12J V . PROTOPOPESCU. Relation entre les operateurs d'albedo et de scat-
tering avec des conditions aux frontieres non-transparentes. C. R. Acad. Sci. Serie 1. 318(1994) 83-86. [13J A. I. PRILEPKO AND N. P . VOLKOV. Inverse problems for deter-
mining the parameters of nonstationary kinetic transport equation from additional information on the traces of the unknown function. Differentsialnye Uravneniya 24(1988) 136-146. [14J M . REED AND B. SIMON. Methods of Modern Mathematical Physics. Vol. 3, Academic Press, New York, 1979. [15J B. SIMON. Existence of the scattering matrix for linearized Boltzmann equation. Comm. Math. Phys. 41(1975) 99-108.
Chapter 14. Inverse scattering and albedo operator
341
[16] P - STEFANOV. Spectral and scattering theory for the linear Boltzmann equation. Math. Nachr. 137(1988) 63-77. [17] J. SYLVESTER AND G . UHLMANN. Global uniqueness for an inverse boundary value problem. Ann. Math. 125(1987) 153-169. [18] J. SYLVESTER AND G. UHLMANN. The Dirichlet to NeumaIlll map and applications. in: Inverse Problems in Partial Differential Equations. SIAM Proc Series List. (1990) 101-139. ed. by D. Colton, R. Ewing and R. Rundell. [19] T. UMEDA. Scattering and spectral theory for the linear Boltzmann operator. J. Math. Kyoto Univ. 24(1984) 208-218. [20] 1. VIDAV. Existence and uniqueness of non-negative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22(1968) 144-155. [21] J. VOIGT. On the existence of the scattering operator for the linear BoltzmaIlll equation. J. Math. Anal. Appl. 58(1977) 541-558. [22] J. VOIGT. Spectral properties of the neutron transport equation. J. Math. Anal. Appl. 106(1985) 140--153.
Index A averages, 30, 245
B Banach lattice, 11, 20, 100
eigenvalue differentiability of leading, 128 isolated , 56 leading, 114, 118, 136 essential spectral radius, 10, 183
C
F
collision frequency , 52, 126 , 300, 302 operator , 59 , 65 , Ill , 125, 281 compactness local , 33 measure of non- , 10, 13 cone positive, 100 critical, 224 criticality eigenvalue problem, 133
factorization technique , 292 of collision operator, 297, 303 Fourier transform of measure, 69 , 73
G group bounded, 270, 280 positive, 269
H
D
hyperplane, 31 , 57 translated, 39, 70
dilation , 61, 68 , 72,74,77, 79 , 84 domination, 20, 38, .59 , 64 , 77 Dunford-Pettis criterion, 82 Dyson-Phillips expansion, 15, 24, 66, 113, 174,203
inverse problems, 245 isotropic models, 163
E
L
eigenelements approximation of leading, 130 leading, 99
limiting absorption principle, 269
I
344
M max-min principle, 157, 161 maximal solution, 223 measure, 30, 32, 51, 59, 66, 70 convolution, 74 Stieljes, 255 truncation, 74 minimal solution , 228 Miyadera perturbation, 174, 199 monotonicity of leading eigenvalue, 124, 126 multigroup transport irreducible, 111
N neutron delayed, 85, 136 nonlinear transport equations, 215 norm additivity, 185, 188, 273
o operator albedo, 311, 339 averaging, 30, 35, 37, 39, 57 compact irreducible, 104 extension, 42, 61 , 67, 72, 80 irreducible, 101 positive, 100 regular , 54, 56-58, 66, 70
Topics in Neutron Transport Tbeory semigroup irreducible, 11, 102 positive, 20, lOCH01 , 188 singular cross sections, 197 transport equations, 197 spectral bound , 19, 56, 102, 106, 146 185 mapping, 25, 93 radius, 104, 119, 191 , 227 spectrum asymptotic, 9, 54 continuous, 87 peripheral, 102 point, 87 real point, 145, 149 reality of point, 164 residual , 87 square root positive, 65, 147 stochastic formulations of neutron transport , 215 strong convex compactness strong topology, 12 subcritical , 224 subcriticality, 117 superconvex scalar mapping, 104 vector valued mapping, 105 weakly , 105 supercritical, 224
R Radon-Nikodym , 62, 84 resolvent irreducible, 102 positive, 186
T
S
transport semigroup irreducible, 109, 137 type essential, 11, 24, 52, 66, 89, 103, 180, 203
selfadjoint positive, 64, 146
W wave operator, 268, 292, 310, 318
Series on Advances in Mathematics for Applied Sciences Editorial Board N. Bellomo Editor-in-Charge Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy E-mail: [email protected]
C. M. Dafermos Lefschetz Center for Dynamical Systems Brown University Providence , RI 02912 USA
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Mathematical Problems in Elasticity ed. R. Russo
Vol. 39
On the Way to Understanding the Time Phenomenon: The Constructions of Time in Natural Science - Part 2. The "Active" Properties of Time According to N. A. Kozyrev ed. A. P. Levich
Vol. 40
Advanced Mathematical Tools in Metrology II eds. P. Ciarlini et al.
Vol. 41
Mathematical Methods in Electromagnetism Linear Theory and Applications by M. Cessenat
Vol. 42
Constrained Control Problems of Discrete Processes by V. N. Phat
Vol. 43
Motor Vehicle Dynamics: Modeling and Simulation byG. Genta
Vol. 44
Microscopic Theory of Condensation in Gases and Plasma by A. L. Itkin and E. G. Kolesnichenko
Vol. 45
Advanced Mathematical Tools in Metrology III eds. P. Cianlini et al.
Vol. 46
Mathematical Topics in Neutron Transport Theory by M. Mokhtar-Kharroubi
New Aspects