The Asymptotic Behavior of Stochastic Anisotropic Navier-Stokes Models with Delays

The aim of this work is to study the pathwise exponential stability of weak solutions of stochastic anisotropic Navier-Stokes equations with delays under the assumptions of slow diffusion in all directions. The existence and uniqueness of stationary solutions to the associated anisotropic Navier-Stokes equations are shown by using the embedding of anisotropic Sobolev spaces. Based on this conclusion, we continue to explore the exponential stability of weak solutions. Our main result provides a relationship among the growth exponents that is suffcient to guarantee the existence, uniqueness and exponential stability of stationary solutions. This is new even for stochastic anisotropic Navier-Stokes equations without delay.