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Communications in Mathematical Sciences
Volume 19 (2021)
Number 4
On entropic solutions to conservation laws coupled with moving bottlenecks
Pages: 919 – 945
DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n4.a3
Authors
Abstract
Moving bottlenecks in road traffic represent an interesting mathematical problem, which can be modeled via coupled PDE-ODE systems. We consider the case of a scalar conservation law modeling the evolution of vehicular traffic and an ODE with discontinuous right-hand side for the bottleneck introduced in [M.L. Delle Monache and P. Goatin, J. Diff. Eqs., 257(11):4015–4029, 2014]. The bottleneck usually corresponds to a slow-moving vehicle influencing the bulk traffic flow via a moving flux pointwise constraint. The definition of solutions requires a special entropy condition selecting non-classical shocks and we prove existence of such solutions for initial data with bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.
Keywords
scalar conservation laws with constraints, PDE-ODE coupled system, wave-front tracking, traffic flow modeling, non-classical shocks
2010 Mathematics Subject Classification
35L65, 90B20
Received 23 October 2019
Accepted 17 November 2020
Published 18 June 2021