Stability of 3d MHD Equations With Anisotropic Dissipation Near a Horizontal Background Magnetic Field

Both physical experiments and mathematical theories indicate that the background magnetic field can stabilize the conducting fluid. Existing results (e.g., [7,20]) show that in the 3D case, either a background magnetic field in a single direction or a magnetic field that satisfies the Diophantine condition can provide a stabilizing mechanism for the fluid. In this paper, we examine a setting that lies between these two cases, specifically a horizontal background magnetic field B₀=(n₁, n₂, 0), n₁, n₂≠ 0. We investigate the 3D incompressible magnetohydrodynamics (MHD) equations with anisotropic dissipation: Fractional viscosity in the velocity field in the x₁-direction and fractional magnetic diffusion in the x₂-direction. Since dissipation acts only in one direction and is fractional, establishing global stability in H³(T³) is nontrivial. We analyze the stabilizing effect of the horizontal magnetic field, combining both the velocity field and the weak dissipation mechanism of the magnetic field, and derive exponential decay in the horizontal directions, which is crucial for the stability proof.