Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks

Martin  Gugat; Alexander  Keimer; Günter  Leugering; Zhiqiang  Wang

doi:10.3934/nhm.2015.10.749

Article Contents

2015, Volume 10, Issue 4: 749-785. Doi: 10.3934/nhm.2015.10.749

This issue Previous Article Optima and equilibria for traffic flow on networks with backward propagating queues Next Article Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics

Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks

1.
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematik, Chair of Applied Mathematics 2, Cauerstraße 11, 91058 Erlangen
2.
School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433

Received: September 30, 2014

Revised: April 30, 2015

Published: November 2015

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Abstract

We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semi-conductor production. Every single commodity is modeled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the dynamics for two commodities. In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered. On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs. For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass. We provide results of regularity, existence and well-posedness of the multi-commodity network model for $L^{p}$-, $BV$- and $W^{1,p}$-data. Moreover, we define an $L^{2}$-tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem.

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Mathematics Subject Classification: 49J20, 93C20, 49N60, 35F61, 35L65, 35L50.

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