Ill-posedness of $2\frac12$D electron MHD

Abstract

We consider the electron magnetohydrodynamics (MHD) in the context where the 3D magnetic field depends only on the two horizontal plane variables. In particular, the magnetic field takes the form $B=\nabla\times (a\vec e_z)+b\vec e_z$ with $a=a(x,y)$ and $b=b(x,y)$. Initial data $(a_0,b_0)$ is constructed in the Sobolev space $H^\beta \times H^{\beta-1}$ with $1<\beta<4$ such that the solution to this electron MHD system either escapes the space or develops norm inflation in $\dot H^\beta \times \dot H^{\beta-1}$.