# DiPerna-Lions flows in $mathsf{RCD}(K,\infty)$ metric measure spaces.

## Dario Trevisan

### (Scuola Normale Superiore, Italy)

Date: Oct 28, 2014, time: 15:25

Abstract. This talk is based on a recent joint work with L. Ambrosio, where a DiPerna-Lions theory of flows of ODEs associated to Sobolev vector fields is established, in a rather general setting. Our aim is to highlight the role played by an abstract superposition principle, linking "Eulerian'' (continuity equations) and "Lagrangian'' (flows of ODEs) viewpoints, and by curvature assumptions on the underlying geometry, providing satisfactory theories, together with non-trivial examples of flows, in the class of $\mathsf{RCD}(K,\infty)$ metric measure spaces.