Asymptotic behaviour of flows on reducible networks

Jacek Banasiak; Proscovia Namayanja; Jacek Banasiak; Proscovia Namayanja

doi:10.3934/nhm.2014.9.197

Networks and Heterogeneous Media

2014, Volume 9, Issue 2: 197-216. doi: 10.3934/nhm.2014.9.197

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Asymptotic behaviour of flows on reducible networks

Jacek Banasiak ^{1
,},
Proscovia Namayanja ^{2
,}

1.
School of Mathematical Sciences, UKZN, Durban
2.
School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041

Received: 01 March 2013 Revised: 01 December 2013
Primary: 47D06, 35F45; Secondary: 05C90, 82C70.

In this paper we extend some of the previous results for a system of transport equations on a closed network. We consider the Cauchy problem for a flow on a reducible network; that is, a network represented by a diagraph which is not strongly connected. In particular, such a network can contain sources and sinks. We prove well-posedness of the problem with generalized Kirchhoff's conditions, which allow for amplification and/or reduction of the flow at the nodes, on such reducible networks with sources but show that the problem becomes ill-posed if the network has a sink. Furthermore, we extend the existing results on the asymptotic periodicity of the flow to such networks. In particular, in contrast to previous papers, we consider networks with acyclic parts and we prove that such parts of the network become depleted in a finite time, an estimate of which is also provided. Finally, we show how to apply these results to open networks where a portion of the flowing material is allowed to leave the network.
- Transport equation,
- network,
- semigroups of operators,
- resolvent positive operators,
- diagraph,
- topological sorting algorithm,
- asymptotic periodicity
Citation: Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks[J]. Networks and Heterogeneous Media, 2014, 9(2): 197-216. doi: 10.3934/nhm.2014.9.197

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Abstract

In this paper we extend some of the previous results for a system of transport equations on a closed network. We consider the Cauchy problem for a flow on a reducible network; that is, a network represented by a diagraph which is not strongly connected. In particular, such a network can contain sources and sinks. We prove well-posedness of the problem with generalized Kirchhoff's conditions, which allow for amplification and/or reduction of the flow at the nodes, on such reducible networks with sources but show that the problem becomes ill-posed if the network has a sink. Furthermore, we extend the existing results on the asymptotic periodicity of the flow to such networks. In particular, in contrast to previous papers, we consider networks with acyclic parts and we prove that such parts of the network become depleted in a finite time, an estimate of which is also provided. Finally, we show how to apply these results to open networks where a portion of the flowing material is allowed to leave the network.

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