# Weak solutions of ideal MHD which do not conserve magnetic helicity

## TL;DR

This paper constructs finite energy weak solutions to the ideal MHD equations that do not conserve magnetic helicity, challenging assumptions about the behavior of such solutions in the zero viscosity limit.

## Contribution

It introduces a Nash-type convex integration method tailored to MHD, demonstrating non-conservation of magnetic helicity in finite energy weak solutions.

## Key findings

- Existence of weak solutions with non-constant magnetic helicity.
- Weak solutions can have finite energy without conserving magnetic helicity.
- Supports the idea that ideal MHD solutions may not be physically realizable as limits of viscous solutions.

## Abstract

We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor's conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.