01325nas a2200133 4500008004100000245008000041210006900121260001000190520088100200100002301081700002101104700001601125856005001141 2014 en d00aRenormalization for autonomous nearly incompressible BV vector fields in 2D0 aRenormalization for autonomous nearly incompressible BV vector f bSISSA3 aGiven a bounded autonomous vector field $b \colon \R^d \to \R^d$, 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. uhttp://urania.sissa.it/xmlui/handle/1963/7483