Singularity formation for the incompressible Hall-MHD equations without resistivity

In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space Hᵐ(R³) for any m > 7/2 . Namely, either the system is locally ill-posed in Hᵐ(R³) , or it is locally well-posed, but there exists an initial data in Hᵐ(R³) , for which the Hᵐ(R³) norm of solution blows-up in finite time if m > 7/2 . In the latter case we choose an axisymmetric initial data u₀(x) = u₀ᵣ(r,z)eᵣ + b0z(r,z)ez and B₀(x) = b0θ(r,z)eθ , and reduce the system to the axisymmetric setting. If the convection term survives sufficiently long time, then the Hall term generates the singularity on the axis of symmetry and we have lim ⁡ _{t→ t⁎}⁡ z ∈ R⁡|∂ z∂ ᵣbθ(r = 0,z)| = ∞ for some t⁎ > 0 , which will also induce a singularity in the velocity field.