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

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 Bogovskiǐ operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.

36 pages