Renormalization for autonomous nearly incompressible BV vector fields in 2D

TL;DR

This paper proves the uniqueness of bounded solutions to the transport equation driven by nearly incompressible BV vector fields in 2D, extending previous steady-state results using a novel partitioning technique and Ambrosio's superposition principle.

Contribution

It introduces a new approach to establish uniqueness for the transport equation with nearly incompressible BV vector fields in 2D, building on and extending prior steady-state results.

Findings

Proved uniqueness of solutions in 2D for nearly incompressible BV vector fields.

Developed a partitioning technique based on level set structure of Lipschitz maps.

Utilized Ambrosio's superposition principle to construct the solution framework.

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}.