JournalsjemsVol. 16, No. 2pp. 201–234

A uniqueness result for the continuity equation in two dimensions

  • Giovanni Alberti

    Università di Pisa, Italy
  • Stefano Bianchini

    SISSA-ISAS, Trieste, Italy
  • Gianluca Crippa

    Universität Basel, Switzerland
A uniqueness result for the continuity equation in two dimensions cover
Download PDF

Abstract

We characterize the autonomous, divergence-free vector fields bb on the plane such that the Cauchy problem for the continuity equation tu+÷(bu)=0\partial_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 ff associated to bb. As a corollary we obtain uniqueness under the assumption that the curl of bb is a measure. This result can be extended to certain non-autonomous vector fields bb 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. 201–234

DOI 10.4171/JEMS/431