Uniqueness and Non-Uniqueness Results for Forced Dyadic MHD Models

We construct non-unique Leray–Hopf solutions for some forced dyadic models for magnetohydrodynamics (MHD) when the intermittency dimension δ is less than 1. Conventionally, the interaction of the velocity and magnetic fields is a major challenge in the context of MHD. However, in the dyadic MHD model scenario, we exploit to our benefit certain symmetries in the interactions of the fields to obtain a non-uniqueness result. In contrast, uniqueness of the Leray–Hopf solution to the dyadic MHD models is established in the case of δ≥ 1 . Analogous results on uniqueness and non-uniqueness of Leray–Hopf solution are also obtained for dyadic models of MHD with fractional diffusion.