A uniqueness result for the continuity equation in two dimensions

Alberti, 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 -