Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

TL;DR

This paper demonstrates the non-uniqueness of weak solutions to 3D magnetohydrodynamic equations, showing they can be close to any smooth divergence-free fields and challenging existing conjectures about ideal limits.

Contribution

It introduces new intermittent flows respecting MHD geometry, enabling the construction of non-unique solutions and analyzing hyper-viscous and hyper-resistive cases up to the Lions exponent.

Findings

Weak solutions do not conserve magnetic helicity.

Weak solutions can approximate any smooth divergence-free fields.

Non-uniqueness persists in hyper-viscous and hyper-resistive MHD up to the critical exponent.

Abstract

We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, we prove that the weak solutions constructed by Beekie-Buckmaster-Vicol [2] for the ideal MHD can be obtained as a strong vanishing viscosity and resistivity limit of a sequence of weak solutions to MHD equations. This shows that, in contrast to the weak ideal limits, Taylor's conjecture does not hold along the vanishing viscosity and resistivity limits. Unlike in the context of the NSE [13] and the ideal MHD [2], new types of velocity and magnetic flows, featuring both the refined spatial and temporal intermittency, are constructed to respect the geometry of MHD and to control the strong viscosity and resistivity. Compatible algebraic structure is derived in the convex integration scheme. More interestingly, the new intermittent flows indeed enable us to prove the aforementioned results for the hyper-viscous and hyper-resistive MHD equations up to the sharp exponent $5/4$, which coincides exactly with the Lions exponent for 3D hyper-viscous NSE.