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].

%B SIAM Journal on Mathematical Analysis %V 48 %P 1-33 %G eng %U https://doi.org/10.1137/15M1007380 %R 10.1137/15M1007380 %0 Report %D 2014 %T Steady nearly incompressible vector elds in 2D: chain rule and renormalization %A Stefano Bianchini %A N.A. Gusev %I SISSA %G en %1 7464 %2 Mathematics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-08-13T07:08:46Z No. of bitstreams: 1 main_stefano(1).pdf: 631783 bytes, checksum: 3ac150a4fb3cb33ebaf5273751dfdf27 (MD5)