Stability threshold of Couette flow for the 2D magnetohydrodynamic equations

The magnetohydrodynamic (MHD) equations admit a physically relevant steady-state solution given by Couette flow combined with a background magnetic field. This paper investigates the nonlinear stability of perturbations around this steady state in an MHD system with partial dissipation. We establish the desired stability when the initial perturbation, measured in terms of the vorticity [Formula: see text] and the current density [Formula: see text], satisfies [Formula: see text], where [Formula: see text] is a suitable constant, [Formula: see text] and [Formula: see text] denotes the viscosity. The exponent [Formula: see text] appears to be optimal and indicates the sharp stability threshold for this MHD system. The proof analyzes the perturbation dynamics over two distinct time intervals and employs the Fourier multiplier method without a change of variables.