Monotone energy stability of magnetohydrodynamics Couette and Hartmann flows

We study the monotone nonlinear energy stability of magnetohydrodynamics plane shear flows, Couette and Hartmann flows . We prove that the least stabilizing perturbations, in the energy norm, are the two-dimensional spanwise perturbations and give some critical Reynolds numbers Re E for some selected Prandtl and Hartmann numbers. This result solves a conjecture given in a recent paper by Falsaperla, Mulone and Perrone, and implies a Squire theorem for nonlinear energy: the less stabilizing perturbations in the energy norm are the twodimensional spanwise perturbations. Moreover, for Reynolds numbers less than Re E there can be no transient energy growth.