Semigroup generation properties of the streaming operator in dependence of the boundary conditions

Abstract

We study the one-dimensional free streaming operator in a slab domain with general boundary conditions described by a linear positive operator A. Under the assumptions that A-' exists and is positive and the free streaming operator T, is resolvent positive, we prove that T, is the infinitesimal generator of a positive strongly continuous semi roup, which is contractive if 11 A ( 1 5 1 and quasi-bounded if71 A 1) > 1 and I( A-' I( 5 1. We give also the mathematical definition of dissipative, conservative and multiplying boundary walls in terms of A, and we show how these properties of the boundary walls can affect the type of the semigroup generated by T,.

Study of the Free Streaming Operator in Slab Geometry in Dependence of the Boundary Conditions

“…Let X G > and X > be the positive cone of X G and X respectively, then we define the following kinds of boundary operators (or boundary walls) (see also [6] for details):…”

“…The well-known result that a semigroup of contraction is generated by ¹ , if # #)1, is obtained in [6] as a particular case. However, in [6], the resolvent operator is only proved to exist and to be positive, but it is not evaluated explicitly.…”

“…Moreover, in [5,6,16] it is shown that conservative boundary conditions lead to a contraction semigroup, whereas multiplying boundary conditions lead to a quasibounded semigroup. The well-known result that a semigroup of contraction is generated by ¹ , if # #)1, is obtained in [6] as a particular case. However, in [6], the resolvent operator is only proved to exist and to be positive, but it is not evaluated explicitly.…”

“…On the other hand, in [7], a more general, but physically reasonable, situation is faced: a three-dimensional bounded convex domain with regular surface, where, as in [6], the boundary conditions are introduced only in the abstract sense, by means of an operator . In [7], the resolvent operator R(z, ¹ ) is expressed in terms of and is proved to be bounded and positive.…”

Some spectral properties of the streaming operator with general boundary conditions

“…The case K = 1 with K 0 has been treated in [2], [7], [13] by approximating from the case K < 1. The case K > 1 has been studied by Borgioli and Totaro [4] for two-dimensional spatial domains and in [3] for three-dimensional spatial domains. We have also studied the case K 1 in [5] by using some geometrical restrictions on X and V .…”