Inserted: 15 aug 2016
Last Updated: 16 oct 2016
Journal: C. R. Math. Acad. Sci. Paris
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.