21 research outputs found
A new approach to transport equations associated to a regular field: trace results and well-posedness
We generalize known results on transport equations associated to a Lipschitz
field on some subspace of endowed with some general
space measure . We provide a new definition of both the transport operator
and the trace measures over the incoming and outgoing parts of generalizing known results from the literature. 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.Comment: 30 page
Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations
We provide a honesty theory of substochastic evolution families in real
abstract state space, extending to an non-autonomous setting the result
obtained for -semigroups in our recent contribution \textit{[On perturbed
substochastic semigroups in abstract state spaces, \textit{Z. Anal. Anwend.}
\textbf{30}, 457--495, 2011]}. The link with the honesty theory of perturbed
substochastic semigroups is established. Several applications to non-autonomous
linear kinetic equations (linear Boltzmann equation and fragmentation equation)
are provided
On perturbed substochastic semigroups in abstract state spaces
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 -spaces or noncommutative 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 -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 spaces. Several
fine properties of Dyson-Phillips expansions are given and a classical
generation theorem by T. Kato is revisited
Integral representation of the linear Boltzmann operator for granular gas dynamics with applications
We investigate the properties of the collision operator associated to the
linear Boltzmann equation for dissipative hard-spheres arising in granular gas
dynamics. We establish that, as in the case of non-dissipative interactions,
the gain collision operator is an integral operator whose kernel is made
explicit. One deduces from this result a complete picture of the spectrum of
the collision operator in an Hilbert space setting, generalizing results from
T. Carleman to granular gases. In the same way, we obtain from this integral
representation of the gain operator that the semigroup in L^1(\R \times \R,\d
\x \otimes \d\v) associated to the linear Boltzmann equation for dissipative
hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic
A new characterization of B-bounded semigroups with application to implicit evolution equations
We consider the one-parameter family of linear operators that A.
Belleni Morante recently introduced and called
B-bounded semigroups. We first determine all the properties
possessed by a couple (A,B) of operators if they generate a B-bounded semigroup (Y(t))t≥0. Then we determine the simplest further property of the couple (A,B) which can assure the existence of a C0-semigroup (T(t))t≥0 such that for all t≥0,f∈D(B) we can write Y(t)f=T(t)Bf. Furthermore, we compare our result with the previous ones and
finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations