Non-uniqueness of weak solutions to the 3D Hall-MHD equations on the plane

TL;DR

This paper demonstrates the non-uniqueness of weak solutions to the 3D Hall-MHD equations on the plane using convex integration, highlighting the existence of multiple solutions with non-trivial magnetic fields.

Contribution

It introduces a novel convex integration approach to construct non-unique weak solutions for the 3D Hall-MHD equations on the plane, including new intermittent flows and analysis of magnetic helicity.

Findings

Weak solutions do not conserve magnetic helicity.

Constructed solutions show non-uniqueness in the specified function space.

Weak solutions can be obtained as limits of vanishing viscosity and resistivity.

Abstract

We prove the non-uniqueness of weak solutions with non-trivial magnetic fields to the 3D Hall-MHD equations on the plane in the space $C^0_t L_x^2$ through the convex integration scheme and by constructing new errors and new intermittent flows. In particular, based on the construction of 3D intermittent flows, we obtain the $2\frac{1}{2}$D Mikado flows through a projection onto the plane. Moreover, we prove that the constructed weak solution do not conserve the magnetic helicity and find that weak solutions of the ideal Hall-MHD equations in $C^{\bar{\beta}}_{t,x}$ ($\bar{\beta}>0$) are the strong vanishing viscosity and resistive limit of weak solutions to the Hall-MHD equations.