Spectra of perturbed semigroups with applications to transport theory
01 May 1970-Journal of Mathematical Analysis and Applications (Academic Press)-Vol. 30, Iss: 2, pp 264-279
About: This article is published in Journal of Mathematical Analysis and Applications.The article was published on 1970-05-01 and is currently open access. It has received 156 citations till now. The article focuses on the topics: Linear transport theory & Neutron transport.
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29 Oct 1999TL;DR: In this paper, Spectral Theory for Semigroups and Generators is used to describe the exponential function of a semigroup and its relation to generators and resolvents.
Abstract: Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.
4,348 citations
TL;DR: In this article, the authors studied the spectral properties of the transport equation and how the diffusion approximation is related to the computation of the critical size, and they showed that when the transport operator is almost conservative, the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate.
Abstract: This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem. Introduction. The computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts. First, the computation of the critical size is much easier for the diffusion approximation than for the original transport equation. Second, for the critical size one can consider a host media X = nXQ with X0 given and 17 a positive number. The problem of the critical size is then reduced to the computation of the parameter zj. It turns out that when the transport operator is almost conservative, the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate. On the other hand the "physical" boundary condition for the diffusion approxi- mation is of the form
287 citations
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10 Feb 1987
TL;DR: In this paper, the authors present a survey of abstract kinetic theory and its application in various areas of physics, chemistry, biology, and engineering, including radiative transfer and rarefied gas dynamics.
Abstract: This monograph is intended to be a reasonably self -contained and fairly complete exposition of rigorous results in abstract kinetic theory. Throughout, abstract kinetic equations refer to (an abstract formulation of) equations which describe transport of particles, momentum, energy, or, indeed, any transportable physical quantity. These include the equations of traditional (neutron) transport theory, radiative transfer, and rarefied gas dynamics, as well as a plethora of additional applications in various areas of physics, chemistry, biology and engineering. The mathematical problems addressed within the monograph deal with existence and uniqueness of solutions of initial-boundary value problems, as well as questions of positivity, continuity, growth, stability, explicit representation of solutions, and equivalence of various formulations of the transport equations under consideration. The reader is assumed to have a certain familiarity with elementary aspects of functional analysis, especially basic semigroup theory, and an effort is made to outline any more specialized topics as they are introduced. Over the past several years there has been substantial progress in developing an abstract mathematical framework for treating linear transport problems. The benefits of such an abstract theory are twofold: (i) a mathematically rigorous basis has been established for a variety of problems which were traditionally treated by somewhat heuristic distribution theory methods; and (ii) the results obtained are applicable to a great variety of disparate kinetic processes. Thus, numerous different systems of integrodifferential equations which model a variety of kinetic processes are themselves modelled by an abstract operator equation on a Hilbert (or Banach) space.
272 citations
01 Jan 1989
TL;DR: The stable, unstable and centre manifolds as discussed by the authors are invariant under the flow relative to a neighbourhood of the equilibrium point and carry the solutions that decay or grow (or neither) at certain rates.
Abstract: When studying the behaviour of a dynamical system in the neighbourhood of an equilibrium point the first step is to construct the stable, unstable and centre manifolds. These are manifolds that are invariant under the flow relative to a neighbourhood of the equilibrium point and carry the solutions that decay or grow (or neither) at certain rates. These ideas have a long history, see for instance Poincare [32] and Hadamard [11]. Sophisticated recent results can be found in Fenichel [7], Hirsch, Pugh and Shub [17] and Kelley [22].
248 citations
TL;DR: In this article, a class of time-periodically perturved evolution equations in a space whose associated Poincar´e map contains a Smale horseshoe is delineated.
Abstract: This paper delineates a class of time-periodically perturved evolution equations in a Banach
space whose associated Poincar´e map contains a Smale horseshoe. This implies that such systems
possess periodic orbits with arbitrarily high period. The method uses techniques originally
due to Melnikov and applies to systems of the form x˙ = f0(x) + "f1(x, t), where x˙ = f0(x)
is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics:
sinusoidally forced vibrations of a buckled beam.
233 citations
References
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01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10
The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field
Zentralblatt MATH, 836
19,846 citations
172 citations
TL;DR: In this article, the authors considered the initial value problem and showed that the decay constant X is an eigenvalue of the Boltzmann operator A. The exact domain of definition of A is dependent on the space of functions where A acts, and will be described in Section 2.
137 citations
112 citations