In this seminar I will present two results regarding the uniqueness (and further properties) for the two-dimensional continuity equation
and the ordinary differential equation in the case when the vector field is bounded, divergence free and satisfies additional conditions on its distributional curl. Such settings appear in a very natural way in various situations, for instance when considering two-dimensional incompressible fluids. I will in particular describe the following two cases:\\
(1) The vector field is time-independent and its curl is a (locally finite) measure (without any sign condition).\\
(2) The vector field is time-dependent and its curl belongs to L^1.\\
Based on joint works with: Giovanni Alberti (Universita' di Pisa), Stefano Bianchini (SISSA Trieste), Francois Bouchut (CNRS &
Universite' Paris-Est-Marne-la-Vallee) and Camillo De Lellis (Universitaet Zuerich).