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 ODEs 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 R-infinity. 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 RCD (K, infinity) metric measure spaces, introduced by Ambrosio, Gigli and Savare [Duke Math. J. 163:7 (2014) 1405-1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.