Seminar Analysis: Dario Trevisan (Scuola Normale Superiore, Pisa, Italy)

Following [1], in this talk we show how to 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 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, recently introduced by Ambrosio, Gigli and Savar ́e, object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results forODE’s under low regularity assumptions on the velocity and in a non smooth context.

References:
[1] L. Ambrosio and D. Trevisan. Well posedness of Lagrangian flows and continuity equations in metric measure spaces. ArXiv e-prints, February 2014.

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