Spectral properties and asymptotic behavior of the linear transport equation
type: Item
instance of: scholarly article
title: Spectral properties and asymptotic behavior of the linear transport equation
zbMATH Open document ID: 0567.45001
DOI: 10.1007/BF01181687
author: Günther Greiner
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173390
review text: The semigroup approach to linear transport problems leads to an abstract Cauchy problem in the Banach lattice \(L^ 1(X\times V)\), where \(X\subseteq {\mathbb{R}}^ 3\) is the configuration space, \(V\subseteq {\mathbb{R}}^ 3\) the velocity space. The phase space \(X\times V\) is equipped with six-dimensional Lebesgue measure. The time-dependent transport equation is given by \((1)\quad (\partial /\partial t)f=A_ 0f- M_{\sigma}f+K_ xf=:Bf,\) \(f(0)=f_ 0\in D(B)\subset L^ 1(X\times V)\) where \(A_ 0\) denotes the differential operator \(- \sum^{3}_{i=1}v_ i(\partial /\partial x_ i)\), \(M_{\sigma}\) is the operator of multiplication by a positive measurable function \(\sigma\) : \(X\times V\to {\mathbb{R}}\) and \(K_ x\) is the integral operator given by \(K_ xf:=\int_{V}x(\cdot,\cdot,v')f(\cdot,v')dv'\), where \(x: X\times V\times V\to {\mathbb{R}}\) is positive and measurable. \(\sigma\) and x are called absorption coefficient and scattering kernel respectively. If the Boltzmann operator \(B:=A_ 0-M_{\sigma}+K_ x\) generates a strongly continuous semigroup \((S(t))_{t\geq 0}\), then the solution of (1) is given by \(f(t):=S(t)f_ 0\). A fundamental problem in linear transport theory is the determination of the asymptotic behavior of the solutions of (1). This problem is intimately related to the spectra of the operators S(t). In this paper we present a complete qualitative solution for the reactor problem and for the multiple scattering problem, assuming only that the absorption coefficient \(\sigma\) and the scattering kernel x are bounded. The methods we use are in the spirit of the article of \textit{G. Birkhoff} [Proc. Symp. Appl. Math. 10, 116-126 (1959; Zbl 0098.234)] that is, besides general spectral theory and the theory of strongly continuous semigroups we mainly use results from the theory of positive linear operators.
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 3905200
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Banach lattice
zbMATH Keywords: time-dependent transport equation
zbMATH Keywords: absorption coefficient
zbMATH Keywords: Boltzmann operator
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: reactor problem
zbMATH Keywords: multiple scattering problem
zbMATH Keywords: positive linear operators
MaRDI profile type: Publication
cites work: On the spectrum of neutron transport equation in finite bodies
cites work: Q3906660
cites work: Über das Spektrum positiver Generatoren
cites work: Compact operators in Banach lattices
cites work: Q3254057
cites work: Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen
cites work: Q3923136
cites work: An asymptotic expansion in the theory of neutron transport
cites work: Spectral methods in linear transport theory
cites work: The spectrum of the multigroup neutron transport operator for bounded spatial domains
cites work: On the spectrum of an unsymmetric operator arising in the transport theory of neutrons
cites work: Fundamental decay mode and asymptotic behaviour of positive semigroups
cites work: Kompaktheit von Integraloperatoren auf Banachverbänden
cites work: Q3882952
cites work: Q4049667
cites work: Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator
cites work: Spectra of perturbed semigroups with applications to transport theory
cites work: A perturbation theorem for the essential spectral radius of strongly continuous semigroups
OpenAlex ID: W2108794751
description: scientific article; zbMATH DE number 3905200