Exponential Mixing of Two‐Dimensional Stochastic Boussinesq Equation for the Magnetohydrodynamic System

This work considers the two‐dimensional Boussinesq equation for a magnetohydrodynamic system driven by white noise. Although it is difficult to observe the long‐time dynamical behavior of an individual solution owing to its chaotic tendency in certain environments, the stationary measure is an effective tool to capture the complex dynamical behavior of fluids in view of statistical dynamics. Thus, we prove the existence and the uniqueness of the stationary measure by examining the strong Feller property of the Markovian semigroup. Furthermore, employing the maximal coupling approach, we establish the exponential mixing of the two‐dimensional Boussinesq equation for a magnetohydrodynamic system.