@article {doi:10.1137/15M1007380,
	title = {Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions},
	journal = {SIAM Journal on Mathematical Analysis},
	volume = {48},
	number = {1},
	year = {2016},
	pages = {1-33},
	abstract = {<p>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{\textquoteright}s superposition principle [4].</p>},
	doi = {10.1137/15M1007380},
	url = {https://doi.org/10.1137/15M1007380},
	author = {Stefano Bianchini and Paolo Bonicatto and N.A. Gusev}
}
