ABSTRACT. We generalize known results on transport equations associated to a Lipschitz field F on some subspace of R N endowed with some general space measure µ. We provide a new definition of both the transport operator and the trace measures over the incoming and outgoing parts of ∂Ω generalizing known results from [9,16]. We also prove the well-posedness of some suitable boundary-value transport problems and describe in full generality the generator of the transport semigroup with no-incoming boundary conditions.

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.

The object of this paper is twofold: In the first part, we unify and extend the recent developments on honesty theory of perturbed substochastic semigroups (on L 1 (µ)-spaces or noncommutative L 1 spaces) to general state spaces; this allows us to capture for instance a honesty theory in preduals of abstract von Neumann algebras or subspaces of duals of abstract C * -algebras. In the second part of the paper, we provide another honesty theory (a semigroup-perturbation approach) independent of the previous resolvent-perturbation approach and show the equivalence of the two approaches. This second viewpoint on honesty is new even in L 1 (µ) spaces. Several fine properties of Dyson-Phillips expansions are given and a classical generation theorem by T. Kato is revisited.

This paper deals with the development of suitable general mathematical structures including a large variety of Boltzmann type models. The contents are organized in three parts. The first part is devoted to modeling the above general framework. The second part to the development of specific models of interest in applied sciences. The third part develops a critical analysis towards research perspectives both on modeling and analytic problems.

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C 0 -semigroup (V H (t)) t 0 in L 1 . With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring (V H (t)) t 0 to be stochastic. We apply these results to examples from kinetic theory.