We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modelled via queues, which form whenever the inflow into a link exceeds its capacity. We answer some rather basic questions about equilibria in this model: in particular uniqueness (in an appropriate sense), and continuity: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions.

To prove these results, we make a surprising connection to another question: whether, assuming constant inflow into the network at the source, do equilibria always eventually settle into a "steady state" where all queue delays change linearly forever more? Cominetti et al. proved this under an assumption that the inflow rate is not larger than the capacity of the network - eventually, queues remain constant forever. We resolve the more general question positively.

(Joint work with Leon Sering and Laura Vargas Koch).