We characterize the autonomous, divergence-free vector fields on the plane such that the Cauchy problem for the continuity equation 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 associated to . As a corollary we obtain uniqueness under the assumption that the curl of is a measure. This result can be extended to certain non-autonomous vector fields with bounded divergence.
Cite this article
Giovanni Alberti, Stefano Bianchini, Gianluca Crippa, A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 201–234DOI 10.4171/JEMS/431