Inserted: 18 jan 2008
Last Updated: 30 jun 2008
Journal: Boll. Unione Mat. Ital. (9)
Proceeding of the conference "XVIII Congresso UMI", Bari, September 2007
In this note we survey some recent results on the well-posedness of the ordinary differential equation with non-Lipschitz vector fields. We introduce the notion of regular Lagrangian flow, which is the right concept of solution in this framework. We present two different approaches to the theory of regular Lagrangian flows. The first one is quite general and is based on the connection with the continuity equation, via the superposition principle. The second one exploits some quantitative a-priori estimates and provides stronger results in the case of Sobolev regularity of the vector field.