# Renormalization and energy conservation for axisymmetric fluid flows

## TL;DR

This paper investigates the behavior of axisymmetric Euler equations with vanishing viscosity, demonstrating energy conservation under certain conditions and establishing the validity of renormalized solutions for vorticity.

## Contribution

It proves energy conservation for axisymmetric Euler solutions with nonnegative vorticity in $L^p$ for $p>3/2$, and confirms these solutions satisfy the vorticity equation in the renormalized sense.

## Key findings

- Energy is conserved for solutions with nonnegative vorticity in $L^p$, $p>3/2$.
- Solutions satisfy the vorticity equation as renormalized solutions.
- Vorticity in $L^p$, $p>1$, leads to well-behaved solutions.

## Abstract

We study vanishing viscosity solutions to the axisymmetric Euler equations with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions. Moreover, we show that the kinetic energy is preserved provided that $p>3/2$ and the vorticity is nonnegative and has finite second moments.