The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier–Stokes equations in unbounded domains

In this paper we consider the existence and uniqueness of weak energy solutions to a stochastic 2-dimensional non-Lipschitz Navier–Stokes equation perturbed by the cylindrical Wiener process W(t)W(t) in a bounded or unbounded domain D with the smooth boundary ∂D   or D=R2D=R2:dX(t)=[−νAX(t)+B(X(t))]dt+f(t,X(t))dt+g(t,X(t))dW(t), where A is the Stokes operator and f, g satisfy the non-Lipschitz condition.