Lorenzo Portinale:
Optimal transport and applications to gradient flows
In this lecture we discuss some of the intriguing connections between the theory of optimal transport and a special class of partial differential equations. In their seminal work, Jordan, Kinderlehrer, and Otto understood that the heat flow equation in Euclidean space can be suitably interpreted as a infinite dimensional gradient-flow equation of the entropy functional with respect to the quadratic Wasserstein distance. One of the challenges of this approach is to provide a rigorous way to handle gradient flows in a general framework such as the one of metric spaces. There are different notions of solutions of a gradient-flow equation in this setting, and starting from the example of the heat flow we will discuss the motivations and the relations between them. Finally, we are going to discuss the advantages that such a variational formulation of certain PDEs brings, including the applications to discrete-to-continuum approximation problems and the generalisation of these ideas to possibly nonlinear and less regular settings.