Renormalization for autonomous nearly incompressible BV vector fields in 2D

Abstract

Given a bounded autonomous vector field $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, 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}.

39 pages, 2 figures

References in corpus (1)

• Lagrangian flows for vector fields with gradient given by a singular integral

Cited by in corpus (3)

• A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients
• Differentiability properties of the flow of 2d autonomous vector fields
• Geometric interpretation of the vanishing Lie Bracket for two-dimensional rough vector fields