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 $x_2$ 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 $H^3\left(R^3\right)$. Furthermore, explicit decay rates in $H^2\left(R^3\right)$ 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.