Calculus of Variations and Geometric Measure Theory
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G. Alberti - S. Bianchini - G. Crippa

A uniqueness result for the continuity equation in two dimensions

created by crippa on 24 Jul 2011
modified by alberti on 12 Feb 2014


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


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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