Growth of the magnetic field in Hall magnetohydrodynamics

While the Hall magnetohydrodynamics (MHD) model has been explored in depth in connection with the dispersive waves relevant in magnetic reconnection, a theoretical study of the mathematical features of this system is lacking. We consider here the boundedness of the solutions of the Hall MHD equations. With Dirichlet boundary conditions the total energy of the system is maintained, and dissipated by diffusion,. but the behaviour of the higher moments of the magnetic field is more complicated. It is found that certain unusual geometries of the initial condition may lead to a blow-up of the L-3-norm of the field. Nevertheless, reasonable assumptions upon the correlation between the size of the magnetic field and. the curvature of field tines imply that the magnetic field remains uniformly bounded.