Abstract
In the present note we discuss in details the Riemann problem for a one-dimensional hyperbolic conservation law subject to a point constraint. We investigate how the regularity of the constraint operator impacts the well--posedness of the problem, namely in the case, relevant for numerical applications, of a discretized exit capacity. We devote particular attention to the case in which the constraint is given by a non--local operator depending on the solution itself. We provide several explicit examples. We also give the detailed proof of some results announced in the paper [Andreianov, Donadello, Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non--local point constraints.
Highlights
1.1 Point constraints in traffic modelingTraffic modeling is an exciting and fast–developing field of research with plentiful applications to real life
While this subject was initially limited to the description and the management of vehicular traffic, we see a growing interest nowadays on different applications as crowd dynamics and bio–mathematics
From the modeling point of view, this may correspond to a narrow exit in crowd modeling, a toll gate in vehicular traffic, a cell membrane in bio-medical modeling
Summary
Traffic modeling is an exciting and fast–developing field of research with plentiful applications to real life. In such situation we say that the point constraint is non–local In this way we obtain crowd and cell membrane dynamics models described by coupled PDE–ODE systems for which the existence and well–posedness of solutions are not trivial matter. Cepolina in [6] prove that the irrational behavior of pedestrians at bottlenecks ends up by reducing the maximal possible outflow This phenomenon, called capacity drop, is related to other effects observed in crowd dynamics, such Faster Is Slower and the Braess’ paradox. The minimal regularity properties to impose on Q in order to achieve well–posedness of solutions are not known at the moment, and they are the object of one of our current research projects
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