26 Mär 2014
15:15  - 16:15

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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