Well-posedness of lagrangian flows and continuity equations in metric measure spaces

Abstract

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∞. 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, ∞) metric measure spaces, introduced by Ambrosio, Gigli and Savaré [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, wellposedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.