Abstract
Since the pioneering work of Jordan, Kinderlehrer, and Otto (JKO), optimal transport has been a powerful tool to study PDEs that can be viewed as gradient flows with respect to the Wasserstein-2 metric. In particular, the optimal transport perspective can be used to create a variational discrete-in-time approximation scheme (the JKO scheme). The JKO scheme has many favorable properties that make it attractive for both theory and numerical simulation. In this talk, I will discuss extensions of the JKO scheme to more general PDEs that cannot be viewed as Wasserstein-2 gradient flows, but can still be evolved using the Wasserstein-2 metric. This includes PDEs featuring mass change such as the Stefan problem and certain tumor growth models, as well as parabolic-hyperbolic equations including the Navier-Stokes equations. This talk is based on joint works with Inwon Kim, Wilfrid Gangbo, Jiajun Tong, and Jaime Marian.