Stability and Large‐Time Behavior for 2D Magnetic Bénard System With Fractional Partial Dissipation in 𝕋×ℝ∗

ABSTRACT In this paper, we focus on the incompressible two‐dimensional magnetic Bénard system with fractional magnetic diffusion and thermal diffusivity in the horizontal direction in the domain . Global stability in and explicit exponential large‐time decay rate of the solution are obtained. Our analysis proceeds by decomposing quantities into a horizontal average and an oscillation component. A pivotal aspect of our analysis is to maximize the hidden dissipation within the equations.