Finite-Resistivity Stabilities of a Sheet Pinch

Abstract

A simple but general approach to obtain the stability criteria is presented through an elementary proof of the uniqueness theorem for the higher order parabolic equations by constructing an energy inequality. The theorem is applied to the isothermal, resistive, viscous and incompressible magnetohy-drodynamical (MHD) equations of a sheet pinch where we reduce the equations into an equation governing the two dimensional vorticity and an equation governing the parallel motion with respect to the acting force. The theorem yields the conditions for the stable parallel motion and for the conservation of vorticity. The result is compared with other works and the effects of viscosity and resistivity are discussed.