Stability for the 2D Anisotropic Magnetohydrodynamic Equations with Only Horizontal Magnetic Diffusion

This paper studies the stability and large-time behavior of perturbations around a large, constant magnetic field in a periodic, infinite channel under specific symmetry constraints. Mathematically, the perturbations are governed by the 2D incompressible magnetohydrodynamic equations with no velocity dissipation and only horizontal magnetic diffusion. This stability result is sharp in the sense that removing this horizontal magnetic diffusion leads to instability. The proof is nontrivial and involves delicate construction of a time-weighted energy functional. Our result rigorously confirms the stabilizing effect of a background magnetic field on electrically conducting fluids.