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

Abstract It has been shown in Jeong and Oh (2019 On the Cauchy problem for the Hall and electron magnetohy-drodynamic equations without resistivity I: illposedness near degenerate stationary solutions Ann. PDE 8 15) that the incompressible and irresistive Hall- and electron-magneto-hydrodynamic (MHD) equations are illposed on flat domains M = R k × T 3 − k for 0 ⩽ k ⩽ 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 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.