Stability and optimal decay for the 3D anisotropic magnetohydrodynamic equations

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 x2$x_2$ 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 H3(R3)$H<^>3(\mathbb {R}<^>3)$. Furthermore, explicit decay rates in H2(R3)$H<^>2(\mathbb {R}<^>3)$ 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.