Communications in Mathematical Sciences

Volume 19 (2021)

Number 4

On entropic solutions to conservation laws coupled with moving bottlenecks

Pages: 919 – 945



Thibault Liard (Facultad de Ingeniería and DeustoTech, Universidad de Deusto, Bilbao, Basque Country, Spain)

Benedetto Piccoli (Department of Mathematical Sciences and CCIB, Rutgers University, Camden, New Jersey, U.S.A.)


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.


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