*Accepted Paper*

**Inserted:** 12 oct 2017

**Last Updated:** 12 oct 2017

**Year:** 2016

**Notes:**

To appear in "Springer Proceedings in Mathematics and Statistics"

**Abstract:**

Given a bounded, autonomous vector field $\mathbf b\colon \mathbb R^d \to \mathbb R^d$, we study the uniqueness of bounded solutions to the initial value problem for the associated transport equation \[ \tag{1} \partial_t u + \mathbf b \cdot \nabla u = 0. \] This problem is related to a conjecture made by A. Bressan, raised studying the well-posedness of a class of hyperbolic conservation laws. Furthermore, from the Lagrangian point of view, this gives insights on the structure of the flow of non-smooth vector fields. In this work we will discuss the two dimensional case and we prove that, if $d = 2$, uniqueness of weak solutions for (1) holds under the assumptions that $\mathbf b$ is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa in ABC1) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem. This is joint work with S. Bianchini and N.A. Gusev BBG.

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