*Accepted Paper*

**Inserted:** 12 sep 2016

**Last Updated:** 27 feb 2017

**Journal:** Bulletin of Mathematical Sciences

**Year:** 2017

**Notes:**

This is an expository paper on the theory of gradient flows (in particular in Wasserstein spaces) and of its latest developments. It has been written on demand, for publication in this Saudi journal that you probably did not know.

Here below is the current (second) version, where I removed many overlaps with my book and added new material.

**Abstract:**

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savar\'e. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise desciption of the Jordan-Kinderleher-Otto scheme, with ideas for the proof of convergence. A discussion of other gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savar\'e, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

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