Given a bounded, autonomous, planar vector field, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation. We prove that uniqueness of weak solutions holds under the assumptions that the vector field is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa) that allows to reduce the transport equation to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem.
In order to perform this program, we use Disintegration Theorem and known results on the structure of level sets of Lipschitz maps: this is done after a suitable localization of the problem, in which we exploit also Ambrosio's superposition principle. This is joint work with S. Bianchini and N.A. Gusev.