# 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

## Abstract

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_t u + \div(bu)=0$ 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 $f$ 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.

## 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