Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

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, for any weak solutions in Ht,xβ˜ to the ideal MHD, where β˜>0, we prove that they are the strong vanishing viscosity and resistivity limit of the 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. Inspired by the works on the NSE [1], ideal MHD [2] and transport equations [3], 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.