A Hamilton-Jacobi approach to junction problems and application to traffic flows^∗

¹ Université Paris-Dauphine, CEREMADE, UMR CNRS 7534, place de Lattre de Tassigny, 75775 Paris Cedex 16, France
imbert@ceremade.dauphine.fr
² École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553, 45 rue d’Ulm, 75230 Paris Cedex 5, France
³ Université Paris-Est, École des Ponts ParisTech, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-La-Vallée Cedex 2, France
monneau@cermics.enpc.fr
⁴ ENSTA ParisTech & INRIA Saclay (Commands INRIA team), 32 boulevard Victor, 75379 Paris Cedex 15, France
Hasnaa.Zidani@ensta.fr

Received: 12 July 2011

Abstract

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.