Solvability of a free-boundary problem describing the traffic flows

Abstract

We study a mathematical model of the vehicle traffc on straight freeways, which describes the traffc flow by means of equations of one dimensional motion of the isobaric viscous gas. The corresponding free boundary problem is studied by means of introduction of Lagrangian coordinates, which render the free boundary stationary. It is proved that the equivalent problem posed in a time-independent domain admits unique local and global in time classical solutions. The proof of the local in time existence is performed with stan- dard methods, to prove the global in time existence the system is reduced to a system of two second-order quasilinear parabolic equations