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

TL;DR

This paper investigates the ill-posedness of 2.5-dimensional electron MHD, showing that solutions can escape the Sobolev space or experience norm inflation, highlighting fundamental issues in the mathematical understanding of the model.

Contribution

It constructs specific initial data demonstrating ill-posedness and norm inflation in the 2.5D electron MHD system, revealing limitations of the Sobolev space framework.

Findings

Solutions can escape the Sobolev space

Norm inflation occurs in the system

Ill-posedness of the 2.5D 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 $\dot H^\beta \times \dot H^{\beta-1}$.