Geodesic
A uniqueness result for the continuity equation in two dimensions
Journal of the European Mathematical Society, Tome 16 (2014) no. 2, pp. 201-234.

Voir la notice de l'article provenant de la source EMS Press

We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation ∂t​u+÷(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.
DOI : 10.4171/jems/431
Classification : 35-XX, 00-XX, 28-XX
Keywords: Continuity equation, transport equation, uniqueness of weak solutions, weak Sard property, disintegration of measures, coarea formula 
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     title = {A uniqueness result for the continuity equation in two dimensions},
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Giovanni Alberti; Stefano Bianchini; Gianluca Crippa. A uniqueness result for the continuity equation in two dimensions. Journal of the European Mathematical Society, Tome 16 (2014) no. 2, pp. 201-234. doi : 10.4171/jems/431. http://geodesic.mathdoc.fr/articles/10.4171/jems/431/

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