Calculus of Variations and Geometric Measure Theory
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L. Caravenna - G. Crippa

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

created by caravenna on 15 Aug 2016
modified by crippa on 16 Oct 2016

[BibTeX]

Accepted Paper

Inserted: 15 aug 2016
Last Updated: 16 oct 2016

Journal: C. R. Math. Acad. Sci. Paris
Year: 2016

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.


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