3D-Streaming Operator with Multiplying Boundary-Conditions: Semigroup Generation Properties

摘要

The present paper concludes a research on the semigroup generation properties of the streaming operator in dependence on the assigned boundary conditions. In two previous papers [4,5], devoted to the same topic in one dimensional slab geometry, results were obtained in the case of multiplying boundary wall 3. In [4] we studied the semigroup generation properties of the streaming operator T with multiplying boundary conditions of Maxwell type. The semigroup generated by T was gained applying a result due to Batty and Robinson (Theorem 2.2.15 in [2]), successively improved by Arendt (Theorem 2.5 in [1]), on the resolvent operator of T , which was expressed as a function of the parameters involved in the boundary conditions. Furthermore and still in one dimensional case, we studied in [5] the properties of the semigroup generated by T with boundary conditions described through a linear positive operator A , representing any possible linear boundary condition (reflection, diffusion, periodic, Maxwell "type, etc.). In [5] it was proved that, in case of ||A||>1 , T generates a positive qLiiasi-bounded semigroup. The well-known result that a semigroup of contraction is generated by T, if ||A|| < 1 , was obtained as a particular case. However, in [5], the resolvent operator was assumed to exist and to be positive, but it was not represented as a function of A .