Nonlinear stability for a simple model of a protoplanetary disc

We study a simple one space dimensional model which has been proposed to describe the behaviour of a protoplanetary disc. The basic model arises from the induction equation for the magnetic field in such a disc. Firstly we analyse the stability and instability thresholds when a shear is imposed and the Hall effect is present. In this situation we employ a classical variational energy method and a transformation variant which achieves an optimal bound. Secondly we also include the effect of ion-slip. A variational energy technique is shown to yield a sharp nonlinear stability threshold.