A uniqueness result for the continuity equation in two dimensions

We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation partial derivative(t)u + div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence.