Some nonlinear stability results for a magnetic induction equation with Hall, ion-slip and shear effects: necessary and sufficient condition

We study the nonlinear stability, with respect to axisymmetric perturbations, of the solution magnetic field to the induction equation for a weakly ionized gas, subjected to an assigned planar velocity field which, in a special case, keeps it in proximity of a gravitational center. In other cases, this velocity field can generate hyperbolic trajectories. Whatever, assuming the presence of Hall and ion-slip effects, we will try to determine how the geometric and kinematic characteristics of the gas stream affect the stability/instability of the magnetic field. Then, we obtain a necessary and sufficient stability condition and estimate the radius of attraction.