Monotone energy stability of magnetohydrodynamics Couette and Hartmann flows

We study the monotone nonlinear energy stability of $\mathit{\text{magnetohydrodynamics}}\mathit{\text{plane}}\mathit{\text{shear}}\mathit{\text{flows,}}\mathit{\text{Couette}}\mathit{\text{and}}\mathit{\text{Hartmann}}\mathit{\text{flows}}$. We prove that the least stabilizing perturbations, in the energy norm, are the two-dimensional spanwise perturbations and give some criti-cal Reynolds numbers Re${}_E$ for some selected Prandtl and Hartmann numbers. This result solves a conjecture given in a recent paper by Falsaperla et al. \cite{FMP.2022} and implies a Squire theorem for nonlinear energy: the less stabilizing perturbations in the $\mathit{\text{energy}}\mathit{\text{norm}}$ are the two-dimensional spanwise perturbations. Moreover, for Reynolds numbers less than Re${}_E$ there can be no transient energy growth.