The semigroup approach to transport processes in networks

On General Transport Equations with Abstract Boundary Conditions. The Case of Divergence Free Force Field

“…To our knowledge, the treatment of transport operators with divergence free with respect to an abstract measure μ vector fields is new. The motivation for studying such problems is to provide an unified treatment of first-order linear problems that should allow to treat in the same formalism transport equations on an open subset of the euclidian space R N (in such a case μ is a restriction of the Lebesgue measure over R N ) and transport equations associated to flows on networks where the measure μ is then supported on graphs (see the recent contributions [20,19,24] on this topic).…”

Explicit formulae for limit periodic flows on networks

“…We note that this theorem is valid under a more general condition than (4), that only involves commensurability of average times of traversing the ergodic cycles, [9,17]. For our purpose we need an explicit expression of (T A (t)) t≥0 , that is the basis of the proof in [1].…”

“…[1,3,5,6,9,13,15] and also the monograph [16]. Here we focus on transport on networks, modelled by a system of first order transport equations on the edges of the digraph representing the network.…”

“…The problem lends itself to a semigroup theoretical approach which, in particular, allows for proving elegant results on the long term behaviour of the flow on the network, see e.g. [1,9,14,17]. The main result of these papers is that, under certain assumptions on the structure of the network and on the speeds of the flow along each edge, the flow is asymptotically periodic.…”

Flows in networks with delay in the vertices

“…The study of such transport processes using semigroup methods has been initiated under the name flows in networks by 14, and since then, many extensions have been considered, such as 7 or 20. See also 8 for a recent survey article.…”