# A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti; Stefano Bianchini; Gianluca Crippa

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 2, page 201-234
- ISSN: 1435-9855

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topAlberti, Giovanni, Bianchini, Stefano, and Crippa, Gianluca. "A uniqueness result for the continuity equation in two dimensions." Journal of the European Mathematical Society 016.2 (2014): 201-234. <http://eudml.org/doc/277674>.

@article{Alberti2014,

abstract = {We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial _tu + \{\frac\{.\}\{\dot\{\}\}\} (bu)=0$ 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.},

author = {Alberti, Giovanni, Bianchini, Stefano, Crippa, Gianluca},

journal = {Journal of the European Mathematical Society},

keywords = {continuity equation; transport equation; uniqueness of weak solutions; weak Sard property; disintegration of measures; coarea formula; transport equation; uniqueness of weak solutions; weak Sard property; disintegration of measures; coarea formula},

language = {eng},

number = {2},

pages = {201-234},

publisher = {European Mathematical Society Publishing House},

title = {A uniqueness result for the continuity equation in two dimensions},

url = {http://eudml.org/doc/277674},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Alberti, Giovanni

AU - Bianchini, Stefano

AU - Crippa, Gianluca

TI - A uniqueness result for the continuity equation in two dimensions

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 2

SP - 201

EP - 234

AB - We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial _tu + {\frac{.}{\dot{}}} (bu)=0$ 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.

LA - eng

KW - continuity equation; transport equation; uniqueness of weak solutions; weak Sard property; disintegration of measures; coarea formula; transport equation; uniqueness of weak solutions; weak Sard property; disintegration of measures; coarea formula

UR - http://eudml.org/doc/277674

ER -

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