Stability of the 2D Boussinesq‐MHD System With Fractional Horizontal Dissipation

ABSTRACT This paper investigates the stability of perturbations near equilibrium states for the two‐dimensional Boussinesq‐magnetohydrodynamics (MHD) system with fractional horizontal dissipation in the domain , where is the one‐dimensional periodic box. The absence of vertical dissipation, combined with the presence of fractional operators, presents significant challenges to the study of stability and long‐time dynamics in the Boussinesq‐MHD system. To address these issues, we leverage the orthogonal decomposition method in conjunction with multiple fractional anisotropic interpolation inequalities and a fractional Poincaré‐type inequality within our analytical framework. We prove the global stability of the system when the initial data are sufficiently small in .