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
.
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
metric measure spaces, introduced by Ambrosio, Gigli and Savaré [DukeMath. 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.
