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Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation

Caravenna, Laura and Crippa, Gianluca. (2016) Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation. Comptes rendus mathematique, 354 (12). pp. 1168-1173.

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Abstract

We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, i.e. transported by a flow of the associated ordinary differential equation. We work in a framework of lack of local integrability of the solution, in which the classical DiPerna-Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration along the unique flow and on a new directional Lipschitz extension lemma, used to construct a large class of test functions in the Lagrangian distributional formulation of the continuity equation.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
UniBasel Contributors:Crippa, Gianluca
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Elsevier Masson for Académie de Sciences
ISSN:1631-073X
Note:Publication type according to Uni Basel Research Database: Journal article
Language:English
Last Modified:20 Jul 2017 08:56
Deposited On:20 Jul 2017 08:53

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