# Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy

## TL;DR

This paper proves that solutions to the 2D Euler equations obtained through the vortex method are Lagrangian and conserve energy when the initial vorticity is in $L^p$ with $p>1$, extending known results to a broader class of solutions.

## Contribution

It demonstrates that vortex method solutions are Lagrangian and energy-conserving for initial vorticity in $L^p$, with $p>1$, broadening the understanding of weak solutions.

## Key findings

- Vortex method solutions are Lagrangian.
- Solutions conserve energy for $p>1$.
- Extends conservation results to broader initial vorticity classes.

## Abstract

We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in $L^p$ with $1\leq p\leq \infty$. Moreover, if $p\geq 3/2$ all weak solutions are conservative. In this work we prove that solutions obtained via the vortex method are Lagrangian, and that they are conservative if $p>1$.