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.

Abstract. We prove the so-called generalized Haff's law yielding the optimal algebraic cooling rate of the temperature of a granular gas described by the homogeneous Boltzmann equation for inelastic interactions with nonconstant restitution coefficient. Our analysis is carried through a careful study of the infinite system of moments of the solution to the Boltzmann equation for granular gases and precise L p estimates in the self-similar variables. In the process, we generalize several results on the Boltzmann collision operator obtained recently for homogeneous granular gases with constant restitution coefficient to a broader class of physical restitution coefficients that depend on the collision impact velocity. This generalization leads to the so-called L 1 -exponential tails theorem for this model.

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.