Explicit transport semigroup associated to abstract boundary conditions
Luisa Arlotti
Conference Publications 2011, 2011(Special): 102-111 doi: 10.3934/proc.2011.2011.102
The linear transport operator associated with abstract bounded boundary conditions, $T_H$, is considered. It is shown that, in some particular cases, a convergent series similar to Dyson-Phillips series can be defined. Sufficient conditions assuring that the sum of this series is a C$_o$-semigroup generated by $T_H$ itself are given.
keywords: Transport equations Boundary conditions Dyson-Phillips series
Transport semigroup associated to positive boundary conditions of unit norm: A Dyson-Phillips approach
Luisa Arlotti Bertrand Lods
Discrete & Continuous Dynamical Systems - B 2014, 19(9): 2739-2766 doi: 10.3934/dcdsb.2014.19.2739
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 $\mathcal{T}_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\mathcal{T}_H$ 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.
keywords: honesty theory. substochastic semigroups Transport equation boundary conditions
Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations
Luisa Arlotti Bertrand Lods Mustapha Mokhtar-Kharroubi
Communications on Pure & Applied Analysis 2014, 13(2): 729-771 doi: 10.3934/cpaa.2014.13.729
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 [On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend. 30, 457--495, 2011]. The link with the honesty theory of perturbed substochastic semigroups is established. Application to non-autonomous linear Boltzmann equation is provided.
keywords: Substochastic semigroups evolution family additive norm linear Boltzmann equation.

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