Ill-posedness of $2\frac{1}{2}$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 ${\dot{H}}^{\beta }\times {\dot{H}}^{\beta -1}$.

arXiv commentinitial data scaling changed; the norm inflation spaces improved to larger range