Abstract time-dependent transport equations
type: Item
instance of: scholarly article
title: Abstract time-dependent transport equations
zbMATH Open document ID: 0657.45007
DOI: 10.1016/0022-247X(87)90252-6
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The following type of time dependent linear initial-boundary value problems is investigated: \[ \partial u/\partial t+\xi \cdot \partial u/\partial x+a\cdot \partial u/\partial \xi +hu+Au=f\quad for\quad x\in \Omega,\quad \xi \in S,\quad 0<t<T; \] u(x,\(\xi\),0)\(=u_ 0(x,\xi)\) for \(x\in \Omega\), \(\xi\in S\); \(u_-(x,\xi,t)=Ku_+(x,\xi,t)+g(x,\xi,t)\) for \((x,\xi)\in B_-\), \(0<t<T\). Here the solution u describing a particle density is to be found, on \(\Omega \times S\times (0,T)\subset {\mathbb{R}}^ n\times {\mathbb{R}}^ n\times {\mathbb{R}}\). The internal sources f and the incident current g are known; \(u_{\pm}\) denote the traces of u on the ``incoming'' and ``outgoing'' parts \(B_{\pm}\) of the boundary, and K is an operator describing the boundary process; the vector field a, the function h and the operator A are also given. There are certain suitable conditions on the data, and in fact the main results are obtained for a more general form of the problem. The problem is treated in \(L_ p\)-setting, \(1\leq p<\infty\), and in the space of measures. It is shown that the problem is well-posed under suitable assumptions, among which \(\| K\| <1\). One of the technical tools is a general Gauss-Green-identity. In the more delicate case of conservative boundary conditions \((\| K\| =1)\) the notion of solution has to be generalized. In the case that the data do not depend on time, the authors show how their results apply to the theory of \(C_ 0\)-semigroups.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 4073701
zbMATH Keywords: abstract time-dependent linear transport equation
zbMATH Keywords: Banach spaces
zbMATH Keywords: space of measures
zbMATH Keywords: kinetic phase space
zbMATH Keywords: dissipativity
zbMATH Keywords: positivity
zbMATH Keywords: evolution equations
zbMATH Keywords: boundary conditions
zbMATH Keywords: \(C_ 0\)-semigroups
zbMATH Keywords: time dependent linear initial-boundary value problems
author: Richard Beals
author: V. Protopopescu
reviewed by: Jürgen Voigt
MaRDI profile type: Publication
cites work: The Cauchy problem for the linearized Boltzmann equation
cites work: Local solutions to the initial and initial boundary value problem for the Boltzmann equation with an external force. I
cites work: Indefinite Sturm-Liouville problems and half-range completeness
cites work: Neutron Transport in a Slab with Moving Boundaries
cites work: Existence, Uniqueness, and Convergence of the Solutions of Models in Kinetic Theory
cites work: On the Boltzmann Equation with External Forces
cites work: Generalized kinetic equations
cites work: Spectral properties and asymptotic behavior of the linear transport equation
cites work: Q3337042
cites work: Spectral methods in linear transport theory
cites work: On the spectrum of an unsymmetric operator arising in the transport theory of neutrons
cites work: Dissipative operators in a Banach space
cites work: Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. I.
cites work: Q5582147
cites work: Q4137618
cites work: Q5535601
cites work: Kinetic Theory and Boundary Conditions for Fluids
cites work: On the classical solutions of the boltzmann equation
cites work: Q4043580
cites work: Q6133405
cites work: Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator
cites work: Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport
full work available at URL: 0022-247x(87)90252-6
OpenAlex ID: W2100220604
description: scientific article; zbMATH DE number 4073701