Semigroup generation properties of the streaming operator in dependence of the boundary conditions
G. Borgioli, Silvia Totaro · Transport Theory and Statistical Physics · 1996
We study the one-dimensional free streaming operator in a slab domain with general boundary conditions described by a linear positive operator Λ. Under the assumptions that Λ−1 exists and is positive and the free streaming operator TΛ is resolvent positive, we prove that TΛ is the infinitesimal generator of a positive strongly continuous semigroup, which is contractive if ∥ Λ ∥ ≤ 1 and quasi-bounded if ∥ Λ > 1 and ∥ Λ−1 ≤ ∥ 1. We give also the mathematical definition of dissipative, conservative and multiplying boundary walls in terms of Λ, and we show how these properties of the boundary walls can affect the type of the semigroup generated by TΛ.