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
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 Wp from the Lp 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.
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
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.