Khalid Latrach

On the essential spectrum of transport operators on L1‐spaces

In a recent article by the first author [J. Math. Phys. 35, 6199–6212 (1994)] the essential spectrum of transport operator was analyzed in Lp‐spaces for p∈(1,+∞). The purpose of the present work is to extend this analysis to the case of L1‐spaces. After establishing preliminary results we define the notion of the weak spectrum which we characterize by means of Fredholm operators. We show, in particular, that in L1‐spaces the weak spectrum is nothing else but the essential spectrum. Using the same techniques as in the above‐mentioned work, we prove the stability of the essential spectrum of a one‐dimensional transport operator with general boundary conditions where an abstract boundary operator relates the incoming and the outgoing fluxes. Sufficient conditions are given in terms of boundary and collision operators, assuring the stability of the essential spectrum. We show also that our results remain valid for neutron transport operators in arbitrary dimension.

Some remarks on the essential spectrum of transport operators with abstract boundary conditions

This article deals with the analysis of the essential spectrum of the one‐dimensional transport operator with general boundary conditions where an abstract boundary operator relates the incoming and the outgoing fluxes. After a complete description of the spectrum of the transport equation with vacuum boundary conditions, sufficient conditions are given in terms of boundary and collision operators assuring the stability of the essential spectrum. The article ends with an analysis of the essential spectrum of the neutron transport operator in arbitrary dimension. Finally, some open problems are indicated.

Spectral Analysis and Generation Results for Streaming Operators with Multiplying Boundary Conditions

This paper deals with the spectral theory of streaming equations for smooth or partly smooth boundary operators. Generation results for muliplying boundary operators in L1-spaces are also given.

EXISTENCE RESULTS FOR A BOUNDARY VALUE PROBLEM ARISING IN GROWING CELL POPULATIONS

The aim of this article is to prove some results regarding the existence of solutions on L1 spaces to a nonlinear boundary value problem derived from a model introduced by Rotenberg (1983) describing the growth of a cell population. Each cell of this population is distinguished by two parameters: its degree of maturity μ and its maturation velocity v. The biological boundary of μ = 0 and μ = a (a > 0) are fixed and tightly coupled through the mitosis. At mitosis, daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. Our proofs, based on topological methods, use essentially the specific properties of weakly compact sets on L1 spaces.

Regularity and time asymptotic behaviour of solutions to transport equations.

We study certain regularity properties of solutions to evolution transport problems which are closely related to spectral properties of transport operators. We consider one-dimensional transport equation for a large class of anisotropic scattering operators and a variety of boundary conditions which cover, in particular, the classical known ones. The analysis uses the geometrical properties of the functional spaces, the results by Gearhart, Slemrod and Wrobel (Theorem 1); and the recent ones by Degond Song. Under adequate assumptions, various descriptions of the large time behaviour of solutions to the associated Cauchy problems are obtained.

Relatively compact-like perturbations, essential spectra and application

The purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. If A denotes a closed densely defined linear operator on a Banach space X, our approach consists principally in considering the class of A-closable operators which, regarded as operators in .X A; X/ (where X A denotes the domain of A equipped with the graph norm), are contained in the set of A-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

REMARKS ON EMBEDDABLE SEMIGROUPS IN GROUPS AND A GENERALIZATION OF SOME CUTHBERT'S RESULTS

Let (U(t))t≥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0 > 0, U(t0) is a Fredholm (resp., semiFredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))t≥0 can be embedded in a C0-group on X. Also we study semigroups which are near the identity in the sense that there exists t0 > 0 such that U(t0)−I ∈ (X), where (X) is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where (X) is replaced by some subsets of the set of polynomially compact perturbations. 2000 Mathematics Subject Classification: 47A53, 47A55, 47D03. 1. Introduction. Let X be a Banach space over the complex field and let (X) denote the Banach algebra of bounded linear operators on X. The subset of all compact operators of (X) is designated by (X) .F orA ∈ (X) ,w e let σ( A), ρ(A), R(λ, A), N(A), and R(A) denote the spectrum, the resolvent set, the resolvent operator, the null space, and the range of A, respectively. The nullity of A, α(A), is defined as the dimension N(A) and the deficiency of A, β(A), is defined as the codimension of R(A) in X. Write

Spectral analysis of transport equations with bounce-back boundary conditions

† ABSTRACT. We investigate the spectral properties of the time-dependent linear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from (1), we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all L p -spaces with 1 < p <1.

Singular one-dimensional transport equations on Lp spaces

We prove the well-posedness of the Cauchy problem governed by a linear mono-energetic singular transport equation (i.e., transport equation with unbounded collision frequency and unbounded collision operator) with specular reflecting and periodic boundary conditions on Lp spaces. The large time behaviour of its solution is also considered. We discuss the compactness properties of the second-order remainder term of the Dyson–Phillips expansion for a large class of singular collision operators. This allows us to evaluate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived.

On an averaging result for transport equations

Abstract In this Note we present some compactness results for transport equations with abstract boundary conditions in bounded geometry.