- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:10
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
16
2014
2
A uniqueness result for the continuity equation in two dimensions
Giovanni
Alberti
Università di Pisa, PISA, ITALY
Stefano
Bianchini
SISSA-ISAS, TRIESTE, ITALY
Gianluca
Crippa
Universität Basel, BASEL, SWITZERLAND
Continuity equation, transport equation, uniqueness of weak solutions, weak Sard property, disintegration of measures, coarea formula
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.
Partial differential equations
General
Measure and integration
201
234
10.4171/JEMS/431
http://www.ems-ph.org/doi/10.4171/JEMS/431