On collisionless transport semigroups with boundary operators of norm one
type: Item
instance of: scholarly article
title: On collisionless transport semigroups with boundary operators of norm one
zbMATH Open document ID: 1171.47035
author: Mustapha Mokhtar-Kharroubi
published in: Journal of Evolution Equations
publication date: 2008-07-08T00:00:00Z
review text: The study of generation results for collisionless streaming operators \(T_H = -(v,\text{grad})\) in subsets of \(\mathbb {R}^n\), with boundary conditions modeled by a positive contraction \(H\) connecting the incoming and outgoing fluxes of particles, goes back to J. Voigt's habilitation thesis (1980). In the context of \({L^1}\)-spaces and norm 1 operators \(H\), covering the case of conservative boundary conditions, some extension \(A \supset T_H\) generates a positive contraction semigroup which need not be stochastic, i.e., mass preserving on the positive cone. Recently, \textit{L. Arlotti} and \textit{B. Lods} [J. Evol. Equ. 5, No.~4, 485--508 (2005; Zbl 1117.47029)] obtained a description of Dom\((A)\). Dealing mainly with conservative \(H\) \((\|H\varphi\|=\|\varphi\|\), \(\varphi\geq0)\), they gave new characterizations of the stochasticity and rederived Voigt's sufficient condition. The present paper treats arbitrary positive \(H\) with \(\|H\|=1\) and, for initial \(f\geq0\), provides an analysis of the total mass carried by individual trajectories \(e^{tA}f\), \( t\geq0\). Among the related established results are sufficient conditions for stochasticity and for \(\overline{T_H}=A\).
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47N55
Mathematics Subject Classification ID: 35F05
Mathematics Subject Classification ID: 82C40
zbMATH DE Number: 5298500
zbMATH Keywords: transport equations
zbMATH Keywords: boundary operators
zbMATH Keywords: substochastic semigroups
zbMATH Keywords: (non)conservative boundary conditions
zbMATH Keywords: mass loss
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: s00028-007-0360-5
OpenAlex ID: W2034748240
DOI: 10.1007/S00028-007-0360-5
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description: scientific article; zbMATH DE number 5298500