Non-uniqueness of Leray weak solutions of the forced MHD equations

TL;DR

This paper demonstrates the non-uniqueness of Leray weak solutions for forced MHD equations by constructing unstable solutions near a special steady state, highlighting limitations of current well-posedness results.

Contribution

It introduces the first unstable background solution for viscous and resistive MHD equations and constructs multiple solutions on the borderline of well-posedness.

Findings

Existence of non-unique Leray weak solutions for forced MHD.

Construction of unstable solutions near a steady state.

Solutions lie on the borderline of known well-posedness theory.

Abstract

In this paper, we exhibit non-uniqueness of Leray weak solutions of the forced magnetohydrodynamic (MHD for short) equations. Similar to the solutions constructed in \cite{ABC2}, we first find a special steady solution of ideal MHD equations whose linear unstability was proved in \cite{Lin}. It is possible to perturb the unstable scenario of ideal MHD to 3D viscous and resistive MHD equations, which can be regarded as the first unstable "background" solution. Our perturbation argument is based on the spectral theoretic approach \cite{Kato}. The second solution we would construct is a trajectory on the unstable manifold associated to the unstable steady solution. It is worth noting that these solutions live precisely on the borderline of the known well-posedness theory.