Global stability of the 2D magnetohydrodynamic equations with horizontal fractional dissipation

This paper focuses on a system of the two-dimensional magnetohydrodynamic equations with horizontal fractional dissipation in the domain Ω = 𝕋 × ℝ with 𝕋 = [ 0 , 1 ] being a periodic box. We study the stability problem on perturbations near any fixed magnetic field 𝐴 = ( 𝐴 1 , 𝐴 2 ) where 𝐴 1 , 𝐴 2 ∈ ℝ . Global stability for this system was proven by Feng et al. [1] in the Sobolev setting H 3 (Ω); however, it seems that one can not establish global stability for the lower regularity solutions to this system by using their method, such as in the Sobolev setting H 2 (Ω). The present work resolves this problem and the global stability for this system in H 2 (Ω) is obtained by utilizing the anisotropic interpolation inequalities and the general strong Poincaré type inequalities, which relaxes the result in [1] . In addition, we also show that the oscillation part of the solution converges to zero in H 1 (Ω) as t → ∞.