Mini-symposium in PDE

Reinhard Illner (University of Victoria), Traffic flow and traffic jams: from kinetic theory to functional differential equations

I will speak on certain kinetic and macroscopic models of traffic flow. After a review of the concept of a fundamental diagram the high-density regime will be considered, and the emergence of macroscopic models with nolocalities will be discussed. Numerical evidence (and real traffic data) suggest that travelling "braking" waves form and propagate in response to trigger events. A traveling wave ansatz for solutions of the macroscopic models leads to an unusual functional differential equation, for which preliminary studies will be shown.

Ben Stephens (University of Washington) Fourth order diffusion with geometric link to second order diffusion

We describe a fourth order family generalizing the linear-mobility thin film equation on R^n. In joint work with R. McCann we derive formally sharp converence rates to self-similarity, using a link to Denzler-McCann's analysis of a second order diffusion. We then show (joint with Matthes, McCann, Savare) that a certain range of nonlinearity allows the obtaining of rigorous results for the fourth-order evolution in 1 dimension.

Robert McCann (University of Toronto), Extremal Doubly Stochastic Measures and Optimal Transportation

Imagine some commodity being produced at various locations and consumed at others. Given the cost per unit mass transported, the optimal transportation problem is to pair consumers with producers so as to minimize total transportation costs. Despite much study, surprisingly little is understood about this problem when the producers and consumers are continuously distributed over smooth manifolds, and optimality is measured against a cost function encoding some geometry of the product space.

This talk will be an introduction to the optimal transportation, its relation to Birkhoff's problem of characterizing of extremality among doubly stochastic measures, and recent progress linking the two. It culminates in the presentation of a criterion for uniqueness of solutions which subsumes all previous criteria, yet which is among the very first to apply to smooth costs on compact manifolds, and only then when the topological type of one of the two underlying manifolds is the sphere.