Limit stationary measures of the stochastic magnetohydrodynamic system in a 3D thin domain

Abstract

This work is concerned with a stochastic magnetohydrodynamic (MHD) system in a 3D thin domain. Although the individual solution may be chaotic in fluid dynamics, the stationary measure is essential to capture complex dynamical behaviors in the view of statistics. We first borrow the α-approximation model to derive the stationary measure of the 3D stochastic MHD system. Then, we further prove that the stationary measure of the system converges weakly to the counterpart of the corresponding 2D stochastic MHD system as the thickness of the thin domain tends to zero.