Time asymptotic behaviour for linear transport equations with abstract boundary conditions in slab geometry

1994 ◽  
Vol 23 (5) ◽  
pp. 633-670 ◽  
Author(s):  
K. Latrach
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Transport Equations ◽  
Slab Geometry ◽  
Linear Transport ◽  
Abstract Boundary

COMPACTNESS RESULTS FOR TRANSPORT EQUATIONS AND APPLICATIONS

2001 ◽  
Vol 11 (07) ◽  
pp. 1181-1202 ◽  
Author(s):  
KHALID LATRACH
Keyword(s):  
Boundary Conditions ◽  
Collision Operator ◽  
Transport Equations ◽  
Boundary Operator ◽  
Systematic Analysis ◽  
Boundary Operators ◽  
Variable Space ◽  
Abstract Boundary ◽  
Two Parameters ◽  
Transport Problems

The goal of this paper is to give a systematic analysis of compactness properties for transport equations with general boundary conditions where an abstract boundary operator relates the incoming and outgoing fluxes. The analysis involves two parameters: The velocity measure and the collision operator. Hence, for a large class of (velocity) measures and under appropriate assumptions on scattering operators compactness results are obtained. Using the positivity (in the lattice sense) and the comparison arguments by Dodds–Fremlin, their converses are derived, and necessary conditions for some remainder term of the Dyson–Phillips expansion to be compact are given. Our results are independent of the properties of the boundary operators and play a crucial role in the understanding of the time asymptotic structure of evolution transport problems. Also, although solutions of transport equations propagate singularities (due to the hyperbolic nature of the operator), they bring the regularity in the variable space (regardless of the boundary operator). We end the paper by applying the obtained results to discuss the existence of solutions to a nonlinear boundary value problem and to describe in detail the various essential spectra of transport operators with abstract boundary conditions.

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

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