Stability and optimal decay for the 3D anisotropic magnetohydrodynamic equations

Abstract This paper investigates the stability problem and large time behavior of solutions to the three‐dimensional magnetohydrodynamic equations with horizontal velocity dissipation and magnetic diffusion only in the direction. By applying the structure of the system, time‐weighted methods, and the method of bootstrapping argument, we prove that any perturbation near the background magnetic field (1, 0, 0) is globally stable in the Sobolev space . Furthermore, explicit decay rates in are obtained. Motivated by the stability of the three‐dimensional Navier–Stokes equations with horizontal dissipation, this paper aims to understand the stability of perturbations near a magnetic background field and reveal the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.