Asymptotic behaviors of stochastic two-dimensional Navier–Stokes equations with finite memory

Abstract

The stochastic 2D Navier–Stokes equations with finite memory are studied. For the differentiable memory function, the almost sure exponential stability of the weak solution is shown by employing a non-negative semimartingale convergence argument. For the nondifferentiable memory function, the exponential stability in mean square for the weak solution is proved by using the differentiability property of the moment function.