A uniqueness result for the continuity equation in two dimensions (Q2441480)

Summary: We characterize the autonomous, divergence-free vector fields \(b\) on the plane such that the Cauchy problem for the continuity equation \(\partial_t u + \mathrm{div}(bu)=0\) 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 non-autonomous vector fields \(b\) with bounded divergence.