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

Caravenna, Laura; Crippa, Gianluca

doi:10.1016/j.crma.2016.10.009

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, that means 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.

Titolo:	Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation
Autori:	CARAVENNA, LAURA Crippa, Gianluca mostra contributor esterni nascondi contributor esterni
Data di pubblicazione:	2016
Rivista:	COMPTES RENDUS MATHEMATIQUE
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, that means 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.
Handle:	http://hdl.handle.net/11577/3199463
Appare nelle tipologie:	01.01 - Articolo in rivista

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