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arXiv · 1412.6387

Renormalization for autonomous nearly incompressible BV vector fields in 2D

Stefano Bianchini·Paolo Bonicatto·Nikolay A. Gusev
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Abstract

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 \cite{BG} (where the \emph{steady} case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in \cite{ABC1}, using the results on the structure of level sets of Lipschitz maps obtained in \cite{ABC2}. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle \cite{ambrosiobv}.

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Stefano Bianchini, Paolo Bonicatto, Nikolay A. Gusev. 2014-12-19. Renormalization for autonomous nearly incompressible BV vector fields in 2D. https://doi.org/10.1137/15m1007380

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