Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space

Abstract

In this paper, we study the axisymmetric Hall magnetohydrodynamics where $u=u^r{e}_r+u^z{e}_z$ and $B=B^{\theta }{e}_{\theta }$. We construct small, smooth, compactly supported, and axisymmetric initial data $(u_0,B_0)$ and prove that there is a unique smooth axisymmetric solution with norm inflation in $H^{\sigma -1}({\mathbb{R}}^3)\times H^{\sigma }({\mathbb{R}}^3)$ for $1<\sigma <7/2$. Moreover, we study the case with fractional dissipation $(-\Delta {)}^{\beta }u$ and $(-\Delta {)}^{\alpha }B$, modifying the initial data slightly and proving that the solution inflates in $H^{\sigma -1}({\mathbb{R}}^3)\times H^{\sigma }({\mathbb{R}}^3)$ for $1<\sigma <7/2-2\max \{\alpha ,\beta \}$ and $0\le \alpha ,\beta <5/4$.

Add the case with dissipation and a priori estimate, together with some minor corrections