The Jordan-Kinderlehrer-Otto scheme in Wasserstein gradient flows for diffusion equations

by Filippo Santambrogio (Université Claude Bernard, Lyon)

Thursday 16 Feb 2023, 15:00 → 16:00 Europe/Rome

Ex-ISEF/Building-Main Lecture Hall (GSSI)

I will start the presentation by recalling what is a gradient flow in the easiest case, i.e. a solution of an equation of the form x'(t)=-grad f(x(t)), and how this equation can be discretized in time as a sequence of optimization problem called minimizing movements. This can be adapted to the case where the evolution of x lives in a functional space and requires to fix a notion of distance on such a space. When we choose the space of probability measures endowed with the Wasserstein distance W_2 the resulting equation is a PDE in the form of a continuity equation, which can become a parabolic diffusion equation for suitable choices of the functional f. This includes the heat equation when f represents the entropy. The discrete optimization scheme used in this case is usually called Jordan-Kinderlehrer-Otto (JKO) scheme.

After a comprehensive introduction to the whole topic I will describe some examples of my current field of research, which consists in finding which properties of the solutions of the continuous-in-time PDEs are shared by the solutions of the JKO scheme (maximum principle, dissipation estimates, …).