Probabilistic representation and uniqueness results for measure-valued solutions of transport equations

The Cauchy problem for a multidimensional linear non-homogeneous transport equation in divergence form is investigated. An explicit and an implicit representation formulas for the unique solution of this transport equation in the case of a regular vector field v are proved. Then, together with a regularizing argument, these formulas are used to obtain a very general probabilistic representation for measure-valued solutions in the case when the initial datum is a measure and the involved vector field is no more regular. but satisfies suitable summability assumptions w.r.t. the solution. Finally, uniqueness results for solutions of the initialvalue problem are derived from the uniqueness of the characteristic curves associated to v through the theory of the probabilistic representation previously developed. (c) 2007 Elsevier Masson SAS. All rights reserved.