# Eulerian and Lagrangian solutions to the continuity and Euler equations with $L^1$ vorticity

## TL;DR

This paper proves uniqueness for continuity equations with $L^1$ vorticity and shows that 2D Euler solutions from vanishing viscosity are renormalized even with low integrability initial data.

## Contribution

It extends the Lagrangian theory to $L^1$ vorticity and confirms renormalization of 2D Euler solutions with minimal initial regularity.

## Key findings

- Uniqueness result for continuity equations with $L^1$ derivatives.
- 2D Euler solutions from vanishing viscosity are renormalized with $L^1$ initial vorticity.

## Abstract

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in \cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \cite{Seis16a} and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in \cite{CrippaSpirito15}.