A perturbation theorem for positive contraction semigroups on L 1-spaces with applications to transport equations and Kolmogorov's differential equations

Abstract

In this paper, a criterion is given for assuring that a linear positive contraction C0-semigrou p defined on an Ll-space is generated by the closure of A + B, A and B being suitable unbounded linear operators. Furthermore, this criterion is applied to the transport equation, Kolmogorov's differential equations, and a transport equation modelling cell growth.AMS subject classifications (1991). 82C70, 45K05.

On transport equations driven by a non‐divergence‐free force field

“…The problem is to construct the smallest possible extension so as not to test (23) on too many unwanted functions. Thus, we briefly recall a particularly efficient extension technique introduced in [12,13] and developed later in [7]. Let E denote the set of -measurable functions defined on X and taking the values in R ∪ {±∞}.…”

On a mathematical model of dust coagulation in interstellar clouds

“…A hint to tackle this problem is suggested by the [12,13], since the unbounded operators A 1 and B satisfy all the fundamental assumptions required for the generation of a strongly continuous semigroup, which are reported in the following equation:…”

New approaches to honesty theory and applications in quantum dynamical semigroups

“…The main idea in Kato's original work in [14] tells us that if A is the generator of a substochastic semigroup on L 1 and B is a positive operator satisfying certain conditions, then there is an extension G of A + B that generates a perturbed substochastic semigroup. Although this theorem is useful as a generation result, with applications in various problems such as birth and death problems, fragmentation problems [14,4] and transport equations [3,24], (see [5, for a survey of the results), our interest in this theorem lies mainly in the honesty theory derived from it.…”

“…Other early results include [24] and [3]. More recently, Voigt and Mokhtar-Kharroubi in [18], introduced a more systematic approach to studying the problem on L 1 , that is, via functionals involving the resolvents of generators.…”

“…The main idea in Kato's original work in [14] tells us that if A is the generator of a substochastic semigroup on L 1 and B is a positive operator satisfying certain conditions, then there is an extension G of A + B that generates a perturbed substochastic semigroup. Although this theorem is useful as a generation result, with applications in various problems such as birth and death problems, fragmentation problems [14,4] and transport equations [3,24], (see [5, Chapters 7-10] for a survey of the results), our interest in this theorem lies mainly in the honesty theory derived from it.Honesty is a property of the perturbed semigroup in Kato's Theorem. We will give the precise technical definition of honesty in Section 2; for now, it suffices to think of honesty theory as the study of the consistency between the perturbed semigroup and the system it describes in the following sense: A substochastic semigroup on L 1 is often used to model the time evolution of some quantity.…”