Mathematics

The theory of flows associated to non-smooth vector fields, developed by Di Perna, Lions, Ambrosio and many other authors, has seen a lot of applications in the study of non-linear partial differential equations. It has been recently extended by Ambrosio and Trevisan to a very general framework including the class of RCD spaces, guaranteeing the existence and uniqueness of suitably generalized flow maps associated with a large class of vector fields.
In joint works with Daniele Semola we investigated the regularity of these flow maps on RCD(K,N) spaces, motivated by the study of geometrical and analytical properties of such spaces. In the Ahlfors regular case, we obtained a result similar to the Euclidean case; we also found a weaker regularity estimate in the general case that has deep applications to the structure theory of RCD(K,N) spaces.
The aim of this talk is twofold:
we first give an insight into the Euclidean theory of Lagrangian flows focusing more on the quantitative side of the theory, pioneered by Crippa and De Lellis. Then we present our result in the case of Ahlfors regular RCD(K,N) spaces, that includes, for instance, Alexandrov and RCD(K,N) non-collapsed spaces.