Asymptotic behavior of solutions to the three-dimensional stochastic Leray-α model

Abstract We consider the three-dimensional stochastic Leray-α model with homogeneous Dirichlet boundary conditions and infinite-dimensional Wiener process. We first study the mean square and pathwise exponential stability of a stationary solution to the model. Then we show that one can stabilize an unstable stationary solution by using a multiplicative Itô noise of sufficient intensity or a linear internal feedback control with support large enough.