arXiv subjects

Luisa Arlotti

Publications and source records attributed to Luisa Arlotti.

6 recordsLinked to original sources

arXiv2018

An $L^{p}$--approach to the well-posedness of transport equations associated to a regular field

We investigate transport equations associated to a Lipschitz field on some subspace of $\mathbb{R}^N$ endowedwith a general measure $μ$ in $L^{p}$-spaces $1 < p <\infty$, extending the results obtained in two previous contributions of the author in the $L^{1}$-context. We notably prove the well-posedness of boundary-value transport problems with a large variety of boundary conditions. New explicit formula for the transport semigroup are in particular given.

math.AP↗

arXiv2013

Transport semigroup associated to positive boundary conditions of unit norm: a Dyson-Phillips approach

We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\T_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\T_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \geq 0}$ as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals $J \subseteq [0,\infty)$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.

math.AP↗

arXiv2013

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 $C_0$-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.

math.AP↗

arXiv2009

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 $L^{1}(μ)$-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 $C^{\ast }$-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.

math.AP↗

arXiv2009

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 $\mathbf{F}$ on some subspace of $\mathbb{R}^N$ 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 $\partial Ω$ 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.

math.AP↗

arXiv2007

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.

math.AP↗