*Published Paper*

**Inserted:** 24 jul 2011

**Last Updated:** 12 feb 2014

**Journal:** Journal of the European Mathematical Society (JEMS)

**Volume:** 16

**Number:** 2

**Pages:** 201-234

**Year:** 2014

**Doi:** 10.4171/JEMS/431

**Abstract:**

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

**Keywords:**
coarea formula, continuity equation, Transport equation, uniqueness of weak solutions, weak Sard property, disintegration of measures

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