Journal of the European Mathematical Society

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Volume 16, Issue 2, 2014, pp. 201–234
DOI: 10.4171/JEMS/431

Published online: 2014-01-23

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti[1], Stefano Bianchini[2] and Gianluca Crippa[3]

(1) Università di Pisa, Italy
(2) SISSA-ISAS, Trieste, Italy
(3) Universität Basel, Switzerland

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.

Keywords: Continuity equation, transport equation, uniqueness of weak solutions, weak Sard property, disintegration of measures, coarea formula

Alberti Giovanni, Bianchini Stefano, Crippa Gianluca: A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. 16 (2014), 201-234. doi: 10.4171/JEMS/431