The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier–Stokes equations driven by Levy processes

Let D   be a bounded or unbounded open domain of 2-dimensional Euclidean space R2R2. If the boundary ∂D=Γ∂D=Γ exists, then we assume that the boundary is smooth. In this paper assuming that the kinematic viscosity ν>0ν>0 is large enough, we discuss the existence and exponential stability of energy solutions to the following 2-dimensional stochastic functional Navier–Stokes equation perturbed by the Levy process:{dX(t)=[νΔX(t)+〈X(t),∇〉X(t)+f(t,X(t))+F(t,Xt)−∇p]dtdX(t)=+g(t,X(t))dW(t)+∫Uk(t,X(t),y)q(dtdy),divX=0in[0,∞)×D, where X(t,x)=φ(t,x)X(t,x)=φ(t,x) is the initial function for x∈Dx∈D and t∈[−r,0]t∈[−r,0] with r>0r>0. It is assumed that f,g,Ff,g,F and k satisfy the Lipschitz condition and the linear growth condition. If there exists the boundary ∂D  , then X(t,x)=0X(t,x)=0 on [0,∞)×∂D[0,∞)×∂D.