On entropic solutions to conservation laws coupled with moving bottlenecks

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

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Received 23 October 2019

Accepted 17 November 2020

Published 18 June 2021