Boundary Value Problems in Abstract Kinetic Theory

A New Approach to Transport Equations Associated to a Regular Field: Trace Results and Well-posedness

“…In particular, we present a new proof of the Green formula clarifying and removing gaps of the proofs in [9,11]. Of course, the boundary condition (1.1b) we treat here is less general than the abstract ones investigated in [9,11] but, as we already mentioned it, the tools we introduce here will allow us to generalize, in a subsequent paper [6], the results of the op. cited by dealing with abstract boundary conditions.…”

“…determination of suitable measures µ ± over the 'incoming' and 'outgoing' parts Γ ± of ∂Ω. We provide here a general construction of these 'trace measures' generalizing, and making more precise, the results of [9,11]. This construction allows us to establish Proposition 2.12 which allows to compute integrals over Ω via integration along the integral curves of F (·) coming from the boundary ∂Ω, and which is free from some restrictive assumptions of op.…”

“…We mention here the seminal works by C. Bardos [8], and by R. Beals and V. Protopopescu [9] (see also [11,14]). Let us however mention that the results of [9,11] introduce restrictive assumptions on the characteristics of the equation. For instance, fields with 'too many' periodic trajectories create serious difficulties.…”

On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions

“…, then we get equality in (14) for the limit function f , giving mass conservation together with uniqueness in the relevant function space (cf [4]- [6], [10], [11], and also Proposition 3.3, chapter 11 in [3]). Remark 2 There holds also in the granular inelastic collision case an H-theorem for a general relative entropy functional…”

“…for describing the neutron distribution in reactor physics, cf. [1]- [3]). In our earlier papers [4]- [6] we have studied the linear Boltzmann equation, both in the angular cut-off case and the infinite range case, for a function f (x, v,t) representing the distribution of particles with mass m undergoing elastic binary collision with other particles with mass m * and with a given (known) distribution function Y (x, v * ).…”