# On the Helicity conservation for the incompressible Euler equations

## TL;DR

This paper studies the regularity and conservation of helicity in weak solutions of the incompressible Euler equations, establishing conditions under which helicity remains constant or exhibits H"older regularity.

## Contribution

It introduces a novel approach treating velocity and curl of velocity as independent functions to analyze helicity conservation and regularity.

## Key findings

- Helicity is conserved under specific regularity conditions.
- Helicity exhibits H"older regularity even when not conserved.
- Conditions relate velocity and curl regularity to helicity behavior.

## Abstract

In this work we investigate the helicity regularity for weak solutions of the incompressible Euler equations. To prove regularity and conservation of the helicity we will threat the velocity $u$ and its $curl\, u$ as two independent functions and we mainly show that the helicity is a constant of motion assuming $u \in L^{2r}_t(C^\theta_x)$ and $curl \,u \in L^{\kappa}_t(W^{\alpha,1}_x)$ where $r,\kappa $ are conjugate H\"older exponents and $2\theta+\alpha \geq 1$. Using the same techniques we also show that the helicity has a suitable H\"older regularity even in the range where it is not necessarily constant.