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Renormalization for autonomous nearly incompressible BV vector fields in 2D

TitleRenormalization for autonomous nearly incompressible BV vector fields in 2D
Publication TypePreprint
2014
AuthorsBianchini, S, Bonicatto, P, Gusev, NA
Document NumberSISSA;67/2014/MATE
InstitutionSISSA

Given 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].

http://urania.sissa.it/xmlui/handle/1963/7483

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