Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit

We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in $L^1_{loc}$) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in $L^1\cap L^\infty$, nonnegative, and with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, e.g. in the Lighthill-Whitham-Richards model for traffic flow) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect $L^1\cap L^\infty \mapsto BV$ for nonlinear scalar conservation laws is intrinsic of the discrete model

Coupling conditions for isothermal gas flow and applications to valves

We consider an isothermal gas flowing through a straight pipe and study the effects of a two-way electronic valve on the flow. The valve is either open or closed according to the pressure gradient and is assumed to act without any time or reaction delay. We first give a notion of coupling solution for the corresponding Riemann problem; then, we highlight and investigate several important properties for the solver, such as coherence, consistence, continuity on initial data and invariant domains. In particular, the notion of coherence introduced here is new and related to commuting behaviors of valves. We provide explicit conditions on the initial data in order that each of these properties is satisfied. The modeling we propose can be easily extended to a very wide class of valves

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 [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.Comment: 19 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1304.628

A macroscopic model for pedestrian flows in panic situations

International audienceIn this paper we present the macroscopic model for pedestrian flows proposed by Colombo and Rosini [10] and show its main properties. In particular, this model is able to properly describe the movements of crowds, even after panic has arisen. Furthermore, it is able to reproduce the so called Braess' paradox for pedestrians. From the mathematical point of view, it provides one of the few examples of non classical shocks motivated by real problems, for which a global existence result is available. Finally, its assumptions were experimentally confirmed by an empirical study of a crowd crush on the Jamarat Bridge in Mina, Saudi Arabia, near Mecca, see [17]

Combined NMDA Inhibitor Use in a Patient With Multisubstance-induced Psychotic Disorder

This document is an Accepted Manuscript reprinted from Journal of Addiction Medicine, Vol. 12 (3): 247-251, May 2018, with permission of Kluwer Law International. Under embargo until 1 May 2019. The Version of Record is available online at DOI: https://doi.org/10.1097/ADM.0000000000000390: Novel psychoactive substance use is a major social concern. Their use may elicit or uncover unpredictably as yet undescribed clinical pictures. We aimed to illustrate a multisubstance use case indistinguishable from paranoid schizophrenia, so to alert clinicians on possibly misdiagnosing substance-induced psychotic disorders. CASE REPORT: We describe a case of a 32-year-old man who started at 18 years with cannabinoids and ketamine, and is currently using N-methyl-D-aspartate (NMDA) antagonists. At age 23, he developed social withdrawal after being assaulted by a stranger, but did not consult psychiatrists until age 26; during this period, he was using internet-purchased methoxetamine and ketamine, and was persecutory, irritable, suspicious, and insomniac and discontinued all received medical prescriptions. He added dextromethorphan to his list of used substances. At age 31, while using phencyclidine, and, for the first time, methoxphenidine, he developed a religious delusion, involving God calling him to reach Him, and the near-death experiences ensured by NMDA antagonists backed his purpose. He received Diagnostic and Statistical Manual of Mental Disorders, 5th Edition diagnosis of multisubstance-induced psychotic disorder and was hospitalized 8 times, 6 of which after visiting the emergency room due to the development of extreme anguish, verbal and physical aggression, and paranoia. He reportedly used methoxphenidine, methoxyphencyclidine, ethylnorketamine, norketamine, and deschlorketamine, to achieve near-death experiences, and eventually to reach God in heavens. CONCLUSIONS: This case points to the need for better control of drugs sold on the internet. It also illustrates that people using NMDA antagonists may present clinical pictures indistinguishable from those of major psychoses and are likely to be misdiagnosed.Peer reviewe

Stability of Transonic Shock Solutions for One-Dimensional Euler-Poisson Equations

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system are investigated. First, a steady transonic shock solution with supersonic backgroumd charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler-Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role