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 related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [4].

1 aBianchini, Stefano1 aBonicatto, Paolo1 aGusev, N.A. uhttps://doi.org/10.1137/15M100738000470nas a2200109 4500008004100000245008400041210006900125260001000194100002300204700001600227856011700243 2014 en d00aSteady nearly incompressible vector elds in 2D: chain rule and renormalization0 aSteady nearly incompressible vector elds in 2D chain rule and re bSISSA1 aBianchini, Stefano1 aGusev, N.A. uhttps://www.math.sissa.it/publication/steady-nearly-incompressible-vector-elds-2d-chain-rule-and-renormalization