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

created by trevisan on 19 Feb 2014
modified on 10 Jun 2017

[BibTeX]

Published Paper

Inserted: 19 feb 2014
Last Updated: 10 jun 2017

Journal: Analysis and PDE
Volume: 7
Number: 5
Pages: 1179-1234
Year: 2014
Doi: 10.2140/apde.2014.7.1179

ArXiv: 1402.4788 PDF
Notes:

Added some comments on the technical condition (7.11), which do not appear in the published version.

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.