Time：10:00-12:00, Monday, April 6 2026
Venue：E5-244
Speaker：Alberto Bressan, Eberly Family Chair Professor, Department of Mathematics Penn State University
Title：Modeling Traffic Flow
Abstract：A mathematical description of traffic flow can be achieved by particle models, in terms of a large number of ODEs describing the position of each car, or by continuum models, in terms of a PDE for the traffic density.
After a general introduction, the talk will cover some recent models of traffic flow, and the new mathematical problems that they generate.
The first part of the talk will focus on a new macroscopic model described by a conservation law with two fluxes, depending on whether the drivers are in "acceleration mode" or in "deceleration mode:. In an unstable regime, a Cauchy problem can have infinitely many entropy-admissible solutions. By introducing a probability measure on the family of all these solutions, one obtains a new stochastic process, whose properties are yet to be studied. Such a model can account for the random creation of stop-and-go waves along a highway.
Vehicular traffic can also be analyzed from the point of view of decision theory. Daily traffic patterns arise as the outcome of the decisions of a large number of drivers, who choose their departure time and route to destination in an optimal way: minimizing a cost for early departure together with a cost for late arrival. In this setting, one can consider globally optimal departure distributions, minimizing the sum of all costs to all drivers, and Nash equilibria, where no driver can lower his own cost by choosing a different departure time. In the second part of the talk, the existence, regularity, and dynamic stability of Nash equilibria will be discussed, together with some open problems.
