Calculus of Variations and Geometric Measure Theory

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


Accepted Paper

Inserted: 15 aug 2016
Last Updated: 16 oct 2016

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


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.