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

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 ${\mathbb{R}}^{\mathrm{\infty }}$. 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 $\mathsf{R}\mathsf{C}\mathsf{D}(K,\mathrm{\infty })$ 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.