On uniqueness of weak solutions to transport equation with non-smooth velocity field

Given a bounded, autonomous vector field $b \colon \mathbb{R}^d \to \mathbb{R}^d$, we study the uniqueness of bounded solutions to the initial value problem for the associated transport equation
\begin{equation}\label{eq:transport} \partial_t u + b \cdot \nabla u = 0. \end{equation}
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 the talk we will discuss the two dimensional case and we prove that, if $d=2$, uniqueness of weak solutions for \eqref{eq:transport} holds under the assumptions that $b$ is of class $\mathrm{BV}$ and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa) that allows to reduce \eqref{eq:transport} 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.