Four Lectures on Mass Transport and Gradient Flows

These lectures are given in the framework of the PDE seminar at the University of Athens, in room A31.
Friday May 9th, 3pm-4.15pm
Friday May 16th, 3pm-4.15pm
Friday May 30th, 3pm-3.50pm
Friday June 6th, 3pm-4.15pm

Program

Lecture 1 The formulations by Monge and Kantorovich. Existence and duality for the Kantorovich problem. Existence of an optimal transport in the strictly convex case. Brenier's Theorem and some consequences.

Lecture 2 Definition and properties of the Wasserstein distances W_p from the L^p transport costs. Link between absolutely continuous curves in the Wasserstein space and the continuity equation.

Lecture 3 Characterization of the geodesics in the Wasserstein spaces. Geodesically convex functionals and applications.

Lecture 4 Short introduction to Gradient Flows ODEs in Rⁿ and in metric spaces. The minimizing movement scheme. How to get some evolution PDEs (heat, Fokker-Plank, porous media, non-local interactions...) as gradient flows for the distance W₂. Structure of the technique.

References

Two classical references are the first book by Cédric Villani Topics in Optimal Transportation (Am. Math. Soc., GSM, 2003) on the general theory, and Gradient Flows in Metric Spaces and in the Space of Probabiliy Measures, by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré (Birkhäuser, 2005). Yet, the approach we use for gradient flows will not exactly be the same as in this book.

A short reference, with almost no proofs, for Lectures 1-3 are the Lecture Notes for a Summer School held in Grenoble in 2009 (to be published now by Cambridge Univ. Press).

I also have a project of a book, an evolution from the above lecture notes and from some notes I wrote for a course (20h) that I gave at Orsay in 2011 and 2012. You can have a look here, it is still in progress.