H\"older continuous dissipative solutions of ideal MHD with nonzero helicity

TL;DR

This paper constructs weak solutions to the 3D ideal MHD equations with Hölder regularity, demonstrating non-conservation of energy and cross helicity while preserving magnetic helicity, using a novel convex integration approach.

Contribution

It introduces a new convex integration scheme that preserves magnetic helicity at each step, enabling the construction of solutions with specific conservation and non-conservation properties.

Findings

Existence of Hölder continuous solutions with nonzero magnetic helicity.

Solutions do not conserve total energy or cross helicity.

First example of solutions where one conserved quantity is preserved while another is not.

Abstract

We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^\alpha$ with $\alpha=1/200$, for which the total energy and the cross helicity (i.e., the so-called Els\"asser energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for H\"older continuous solutions, is nonzero. The construction, which works both on the torus $\mathbb{T}^3$ and on $\mathbb{R}^3$ with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations.