Half Strong Ill-Posedness of $2\frac{1}{2}$D Electron Magnetohydrodynamics with Fractional Resistivity

Abstract

We study the $2\frac{1}{2}$D electron magnetohydrodynamics (MHD): the electron MHD system that has $3$D magnetic field but is independent of $z$-variable. We establish a "half" strong ill-posedness result in $2\frac{1}{2}$D electron MHD with fractional resistivity $(-\hat{I}”{)}^{\hat{I}}\pm$ in the supercritical Sobolev space $H^2\times H^{{\hat{I}}^2-1}$ for $3<{\hat{I}}^2<4-2\hat{I}\pm$. Specifically, we construct small initial data $(a_0,b_0)$ whose solution develops a norm inflation in $a$ but the norm of $b$ remains small.