Relax the magnetic-field threshold for nonlinear stability

Establish nonlinear stability of the two-dimensional MHD Couette flow with vertical viscosity and vertical magnetic diffusion under a lower bound on the background horizontal magnetic-field strength closer to the linear stability threshold |β|>1/2, by developing a more refined symmetrizer than the one used in the present energy framework.

Background

The nonlinear stability theorem requires the strong-field condition |β|>7π+1, together with a compatibility condition involving the viscosity, magnetic diffusivity, and |β|. The authors explain that these numerical restrictions arise from the particular symmetrized energy and multiplier estimates used in the proof, rather than from a known sharp physical stability threshold.

The inviscid linearized problem is stable in the regime |β|>1/2, while the nonlinear theorem only treats substantially larger magnetic-field strengths. The unresolved problem is therefore to determine whether a refined symmetrizer can reduce the nonlinear field-strength requirement toward the linear threshold |β|>1/2.