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 R 2 . If the boundary ∂ D = Γ exists, then we assume that the boundary is smooth. In this paper assuming that the kinematic viscosity ν > 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: { d X ( t ) = [ ν Δ X ( t ) + 〈 X ( t ) , ∇ 〉 X ( t ) + f ( t , X ( t ) ) + F ( t , X t ) − ∇ p ] d t d X ( t ) = + g ( t , X ( t ) ) d W ( t ) + ∫ U k ( t , X ( t ) , y ) q ( d t d y ) , d i v X = 0 in [ 0 , ∞ ) × D , where X ( t , x ) = φ ( t , x ) is the initial function for x ∈ D and t ∈ [ − r , 0 ] with r > 0 . It is assumed that f , g , F and k satisfy the Lipschitz condition and the linear growth condition. If there exists the boundary ∂D, then X ( t , x ) = 0 on [ 0 , ∞ ) × ∂ D .