Riemann problems with non--local point constraints and capacity drop

Boris Andreianov (1) , Carlotta Donadello (1) , Ulrich Razafison (1) , Massimiliano D. Rosini (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 ICM

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 [Andreainov, Donadello, Rosini, "Crowd dynamics and conservation laws with non--local point 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.

Mots clés: capacity drop; Riemann problem; non--local constrained hyperbolic PDE's; loss of self--similarity; loss of uniqueness; crowd dynamics; capacity drop.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini. Riemann problems with non--local point constraints and capacity drop. Modeling with Measures, Aug 2013, Leiden, Netherlands. pp. 259-278, ⟨10.3934/mbe.2015.12.259⟩. ⟨hal-00959974⟩