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where X is the velocity field of the fluid, ν > 0 is the kinematic viscosity, p is the associated hydrostatic pressure, f is a non-delayed external force field, g(X(t − ρ(t))) is another external force field with delay, σ(t, X(t− δ(t)))dW(t) + ∫ Z k(t, X(t− γ(t)), z)η̃(dt, dz) is a random external force field with delays, where W is an infinite dimensional Wiener process and η̃ is a compensated time homogeneous Poisson random measure, ρ, δ, γ : R+ → [0, r] are continuous functions, D is a regular open bounded domain of R2 with boundary ∂D, X0 is the initial velocity, φ is the initial datum in the interval [−r, 0]. Navier-Stokes equations are the fundamental model of the fluid mechanics and turbulence. These equations have been the object of numerous works (see, for instance [1,2] and the references therein), even with unbounded domains (see [2,3]) since the first paper of Leray was published in 1933 [4]. Beside, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of Navier-Stokes equations. To our best knowledge, the theory of stochastic Navier-Stokes equations apparently has its roots in the 1959 edition of the Landau and Lifshitz [5] and the first work on the stochastic Navier-Stokes equations written from the mathematical point of view is the paper [6]. We mention here the works ([7–10] and the references cited therein) concerning the two-dimensional (2D in short) stochastic Navier-Stokes equation. Partial differential equations with memory (e.g. delay) have attracted great interest due to their applications in describing many sophisticated dynamical systems in physics, chemistry, biology, economics and social sciences. On this matter, we refer the reader to [11–16] and the references therein. In the past few years, many papers have studied the Navier-Stokes equations with a forcing term which contains some hereditary characteristics (see, for instance [17–19] and the references therein). In addition, the long-time behavior and exponential stability of the Navier-Stokes equations is an interesting and challenging problem, since it can provide useful information on the future evolution of the system (see, [20,21]). On the exponential behavior for stochastic 2D Navier-Stokes equations with variable delay, we refer the reader to [22–24], and

Stability of stochastic 2D Navier-Stokes equations with memory and Poisson jumps