Singularity formation for the incompressible Hall-MHD equations without resistivity

Abstract

In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space $H^m({\mathbb{R}}^3)$ for any $m>\frac{7}{2}$. Namely, either the system is locally ill-posed in $H^m({\mathbb{R}}^3)$, or it is locally well-posed, but there exists an initial data in $H^m({\mathbb{R}}^3)$, for which the $H^m({\mathbb{R}}^3)$ norm of solution blows-up in finite time if $m>\frac{7}{2}$. In the latter case we choose an axisymmetric initial data $u_0(x)=u_{0r}(r,z)e_r+b_{0z}(r,z)e_z$ and $B_0(x)=b_{0\theta }(r,z)e_{\theta }$, 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 $\lim {\sup }_{t\to t_{⁎}}{\sup }_{z\in \mathbb{R}}|{\partial }_z{\partial }_r{b}_{\theta }(r=0,z)|=\infty$ for some $t_{⁎}>0$, which will also induce a singularity in the velocity field.