Substochastic semigroups for transport equations with conservative boundary conditions

Abstract

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C 0 -semigroup (V H (t)) t 0 in L 1 . With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring (V H (t)) t 0 to be stochastic. We apply these results to examples from kinetic theory.

On transport equations driven by a non‐divergence‐free force field

“…The strategy is the same as that employed in [1, Section 3] for divergence-free transport fields with the only difference being the expression for the boundary measures ± . Hence, we adopt here the same general assumptions on the measure as in [1].…”

“…[1]). With these considerations, we can represent, up to a set of zero measure, the phase space X T as…”

“…In contrast with our previous work [1], the field F is not assumed to be divergence free with respect to . Precisely, we define the divergence of F with respect to according to the following assumption.…”

“…In [1], we studied a general theory of initial boundary-value problems for transport equations with divergence-free external field. As in most earlier works (e.g.…”

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

“…µ(T t A) = µ(A) for any measurable subset A ⊂ R N and any t ∈ R. where F = (F 1 , F 2 , F 3 ) is a time independent force field over D×R 3 satisfying Assumption 1 and (1.2). The free transport case, investigated in [16,4], corresponds to F = 0. The existence of solution to the transport equation (1.1a) is a classical matter when considering the whole space Ω = R N .…”