In the nineteenth century, the transport equation became the first model of fluid dynamics to be set on firm foundations by the method of characteristics valid only for smooth vector fields and smooth initial datum. In the first half of the twentieth century, the notion of weak solution emerged motivated by the question of existence of solutions for non-linear evolution equation. For the transport equation, the method of characteristics carries through verbatim to weak solutions only when the vector field is smooth. But, when the vector field is rough, the theory of weak solutions for the transport equation has a rich history.
In the late twentieth century, DiPerna and Lions established well-posedness for bounded weak solutions when the vector field is Sobolev, and a measure-theoretic uniqueness for the flow of Sobolev vector fields. Recent works have used convex integration to construct non-unique trajectories of Sobolev vector fields for almost every initial datum thereby showing that the measure-theoretic notion of uniqueness for the flow put forward by DiPerna and Lions is strictly weaker than point wise uniqueness of trajectories.
Twenty years ago, Ambrosio proved well-posedness for bounded weak solutions when the vector field is of bounded variation. An example of Depauw then showed that this result is sharp: uniqueness of bounded weak solution is not to be expected for a general bounded vector field. But a weaker notion of uniqueness is still within reach for the Depauw vector field, and in fact for a whole class of vector field comprising it. A corresponding unique flow, stochastic in nature, also exists for such vector field. However for merely bounded vector fields, this weaker notion of uniqueness fails over a dense set.