Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow

In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow. We also give a stability result with respect to various parameter

Nonlocal Crowd Dynamics Models for several Populations

This paper develops the basic analytical theory related to some recently introduced crowd dynamics models. Where well posedness was known only locally in time, it is here extended to all of \reali^+. The results on the stability with respect to the equations are improved. Moreover, here the case of several populations is considered, obtaining the well posedness of systems of multi-D non-local conservation laws. The basic analytical tools are provided by the classical Kruzkov theory of scalar conservation laws in several space dimensions

Control of the Continuity Equation with a Non Local Flow

This paper focuses on the optimal control of weak (i.e. in general non smooth) solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of a class of equations comprising these models. In particular, we prove the differentiability of solutions with respect to the initial datum and characterize its derivative. A necessary condition for the optimality of suitable integral functionals then follows

An Analytical Framework to Describe the Interactions Between Individuals and a Continuum

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheeps. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided

Conservation laws with a non-local flow application to pedestrian traffic

In this note, we introduce some models of pedestrian traffic and prove existence and uniqueness of solutions for these models

Étude de différents aspects des EDP hyperboliques : persistance d'onde de choc dans la dynamique des fluides compressibles, modélisation du trafic routier, stabilité des lois de conservation scalaires.

In this work, we study hyperbolic systems of balance laws. The first part is devoted to compressible fluid dynamics, and particularly to the lifespan of smooth or piecewise smooth solutions. After presenting the state of art, we show an extension to more general gases of a theorem by Grassin. We also study shock waves solutions: first, we extend T. T. Li's approach to estimate the time of existence in the isentropic spherical case; second, we develop Whitham's ideas to obtain an approximated equation satisfied by the discontinuity surface. In the second part, we set up a new model for a roundabout. This leads us to study a multi-class extension of the macroscopic Lighthill-Whitham-Richards' model. We study the traffic on an infinite road, with some points of junction. We distinguish vehicles according to their origin and destination and add some boundary conditions at the junctions. We obtain existence and uniqueness of a weak entropy solution for the Riemann problem. As a complement, we provide numerical simulations that exhibit solutions with a long time of existence. Finally, the Cauchy problem is tackled by the front tracking method. In the last part, we are interested in scalar hyperbolic balance laws. The first question addressed is the control of the total variation and the stability of entropy solutions with respect to flow and source. With this result, we can study equations with non-local flow, which do not fit into the framework of classical theorems. We show here that these kinds of equations are well posed and we show the Gâteaux-differentiability with respect to initial conditions, which is important to characterize maxima or minima of a given cost functional.On étudie dans ce travail des systèmes de lois de conservation hyperboliques. La première partie étudie le temps d'existence des solutions régulières et régulières par morceaux de la dynamique des fluides compressibles. Après avoir présenté l'état de l'art en matière de solutions régulières, on montre une extension d'un théorème de Grassin à des gaz de Van der Waals. On étudie ensuite les solutions ondes de chocs : on poursuit l'approche de T. T. Li pour estimer leur temps d'existence dans le cas isentropique à symétrie sphérique, et l'approche de Whitham afin d'obtenir une équation approchée vérifiée par la surface de discontinuité. Dans une deuxième partie, motivée par la modélisation d'un rond-point en trafic routier, on étudie une extension multi-classe du modèle macroscopique de Lighthill-Whitham-Richards sur une route infinie avec des jonctions. On différencie les véhicules selon leur origine et leur destination et on introduit des conditions aux bord adaptées au niveau des jonctions. On obtient existence et unicité d'une solution au problème de Riemann pour ce modèle. Des simulations numériques attestent que les solutions obtenues existent en temps long. On aborde enfin le problème de Cauchy par la méthode de front tracking. La dernière partie concerne les lois de conservation scalaires. La première question abordée est le contrôle de la variation totale de la solution et la stabilité des solutions faibles entropiques par rapport au flux et à la source. Ce résultat nous permet d'étudier des équations avec flux non-local. Une fois établi leur caractère bien posé, on montre la Gâteaux-différentiabilité du semi-groupe obtenu par rapport aux conditions initiales