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

TL;DR

This paper demonstrates that the 2.5D electron MHD system with fractional resistivity exhibits a form of ill-posedness, where small initial data can lead to rapid norm inflation in solutions, indicating instability in certain Sobolev spaces.

Contribution

It establishes a 'half' strong ill-posedness result for 2.5D electron MHD with fractional resistivity in supercritical Sobolev spaces, a novel instability analysis.

Findings

Constructed small initial data leading to norm inflation.

Proved ill-posedness in supercritical Sobolev spaces.

Identified conditions for instability in the system.

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 $(-\Delta)^\alpha$ in the supercritical Sobolev space $H^{\beta}\times H^{\beta-1}$ for $3<\beta<4-2\alpha$. 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.