Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space

Abstract

In this paper, we study the axisymmetric Hall magnetohydrodynamics where $u = u^re_r+u^ze_z$ and $B = B^θe_θ$. 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 $H^{σ-1}(\mathbb R^3) \times H^σ(\mathbb R^3)$ for $1<σ<7/2$. Moreover, we study the case with fractional dissipation $(-Δ)^βu$ and $(-Δ)^αB$, modifying the initial data slightly and proving that the solution inflates in $H^{σ-1}(\mathbb R^3) \times H^σ(\mathbb R^3)$ for $1<σ<7/2-2\max\{α,β\}$ and $ 0 \le α, β< 5/4$.