Riemann problems with non-local point constraints and capacity drop

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.