Abstract kinetic equations relevant to supercritical media
type: Item
instance of: scholarly article
title: Abstract kinetic equations relevant to supercritical media
zbMATH Open document ID: 0584.47050
DOI: 10.1016/0022-1236(84)90092-2
author: William Greenberg
author: C. van der Mee
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: The abstract Hilbert space equation \((Tf)'(x)=-(Af)(x)\), \(x\geq 0\) with partial range boundary condition \((Q_+f)(0)_+\) is of considerable importance for the linear transport theory. This equation has mainly be studied under the conditions that T is injective and selfadjoint and that A is Fredholm and positive (and quite often injective). The restriction to positive A excludes many applications. In the present paper T is assumed to be bounded, injective and selfadjoint and A to be selfadjoint and Fredholm with finite negative part. The completion \(H_ A\) of the domain D(A) with respect to the form \((x,y)_ A:=(Ax,y)\) is an indefinite inner product space with finite-dimensional nonpositive part. Under the additional requirement that A be T-regular (basically, this means that the spectral resolution of \(T^{-1}A\) in the space \(H_ A\) is at no point in \({\mathbb{R}}\) unbounded) the authors derive a complete existence-and-uniqueness theory for the abstract boundary value problem. To do this, they reduce it to two subproblems: (1) a half-space-problem of the same type with strictly positive \(\tilde A\) on a finite-codimensional subspace and (2) an evolution equation on a finite-dimensional subspace Z. On the first problem many results are available; the problem (2) is solved by analysing the space Z with the indefinite inner product \([u,v]:=(Tu,v)\). The measure of nonsolvability and non-uniqueness of the problem can be expressed through the sign characteristic of \(T^{- 1}A|_ Z\) with respect to the form \([\cdot,\cdot].\) The results are applied to the one-speed neutron- and to the symmetric multigroup transport equation.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 82C70
Mathematics Subject Classification ID: 82D45
Mathematics Subject Classification ID: 47A50
zbMATH DE Number: 3935827
zbMATH Keywords: supercritical media
zbMATH Keywords: abstract Hilbert space equation
zbMATH Keywords: partial range boundary condition
zbMATH Keywords: linear transport theory
zbMATH Keywords: injective and selfadjoint
zbMATH Keywords: Fredholm and positive
zbMATH Keywords: spectral resolution
zbMATH Keywords: complete existence-and- uniqueness theory for the abstract boundary value problem
zbMATH Keywords: half-space- problem
zbMATH Keywords: evolution equation
zbMATH Keywords: indefinite inner product
zbMATH Keywords: measure of nonsolvability and non-uniqueness
zbMATH Keywords: results are applied to the one-speed neutron- and to the symmetric multigroup transport equation
reviewed by: G. Philip. A. Thijsse
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90092-2
OpenAlex ID: W2009656796
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description: scientific article; zbMATH DE number 3935827