The Follow-The-Leader model without a leader: An infinite-dimensional Cauchy problem

We introduce a Follow-The-Leader model where all drivers are indistinguishable, in the sense that each driver has a role and a behavior depending on the driver immediately in front, but none of them has the a-priori privilege to drive at maximal speed. We prove that the resulting Cauchy Problem with infinitely many differential equations admits a global unique solution. The total variation of the discrete density being uniformly bounded, as the proper length of each vehicle vanishes the infinite microscopic model converges to the macroscopic LWR model based on a first order PDE. Finally, the case of traffic flow with monotone density is discussed, with applications to real situations.