On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space

TL;DR

This paper proves strong illposedness of the incompressible, resistivity-free Hall- and electron-MHD equations in three-dimensional space, extending previous results by removing simplifying assumptions and constructing degenerating wave packets.

Contribution

It demonstrates the illposedness of these MHD equations without the independence assumption, using advanced analytical tools and wave packet constructions.

Findings

Strong illposedness for compactly supported data in D

Construction of degenerating wave packets around axisymmetric magnetic fields

Extension of previous results to more general data without independence assumptions

Abstract

It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M = \mathbb{R}^k \times \mathbb{T}^{3-k}$ for $0 \le k \le 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^3$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovski\v{i} operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.