In this talk, we will show the non-uniqueness of weak solutions to 3D generalized MHD equations, up to the sharp Lions exponent. The constructed weak solutions do not conserve the magnetic helicity and can be close to the given smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, we prove that the weak solutions constructed by Beekie-Buckmaster-Vicol for the ideal MHD can be obtained as a strong vanishing viscosity and resistivity limit of a sequence of weak solutions to the generalized MHD. In particular, this shows that, in contrast to the weak ideal limits, Taylor’s conjecture does not hold along the vanishing viscosity and resistivity limits of weak solutions to the generalized MHD. The talk is based on the work in joint with Yachun Li and Zirong Zeng.

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