Non-unique solutions for electron MHD

TL;DR

This paper demonstrates the non-uniqueness of weak solutions for the electron MHD equations on a 3D torus, including solutions without resistivity, by constructing solutions arbitrarily close to any given smooth vector field.

Contribution

It introduces a method to construct non-unique weak solutions to electron MHD, including non-Leray-Hopf solutions, expanding understanding of solution behavior under hyper-resistivity.

Findings

Non-uniqueness of weak solutions for electron MHD with hyper-resistivity.

Existence of weak solutions without resistivity.

Construction of solutions arbitrarily close to any smooth vector field.

Abstract

We consider the electron magnetohydrodynamics (MHD) equation on the 3D torus $\mathbb T^3$. For a given smooth vector field $H$ with zero mean and zero divergence, we can construct a weak solution $B$ to the electron MHD in the space $L^\gamma_tW^{1,p}_x$ for appropriate $(\gamma, p)$ such that $B$ is arbitrarily close to $H$ in this space. The parameters $\gamma$ and $p$ depend on the resistivity. As a consequence, non-uniqueness of weak solutions is obtained for the electron MHD with hyper-resistivity. In particular, non-Leray-Hopf solutions can be constructed. As a byproduct, we also show the existence of weak solutions to the electron MHD without resistivity.