Caltech/UCLA/USC Joint Analysis Seminar

In 2001 F. Otto discovered a (nowadays well-known) relationship between the continuity equation and gradient flows with respect to the 2-Wasserstein metric. This connection provides a convenient description of many new and classical models and PDEs including Keller-Segel and Fokker-Planck as well as models of first-order collective dynamics. 

I am going to present a recent work (joint with David Poyato), wherein we introduce the so-called fibered 2-Wasserstein metric (which admits only transportation along fibers controlled by a prescribed probabilistic distribution) and explore its applicability in gradient flows. Based on such a metric, we develop the notion of heterogeneous gradient flows, and prove that they are equivalent to solutions of parameterized continuity equations. Lastly, I will present a collection of applications ranging from mixtures of fluids, to multispecies models of collective dynamics, and to (the essential) applications in alignment models.