# Well posedness of Lagrangian flows and continuity equations in metric measure spaces

**published**
**year:** 2014

**journal:** Analysis and PDE

**abstract:** We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$.
When specialized to the setting of Euclidean or infinite dimensional (e.g.\ Gaussian) spaces, large parts of previously known results are recovered at once.
Moreover, the class of ${\sf RCD}(K,\infty)$ metric measure spaces object of extensive recent research fits into our framework.
Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

The paper is available on the
cvgmt preprint server.