Nonlinear colored noise in incompressible Navier-Stokes equations on two-dimensional unbounded domains

In this paper we consider a randomly-forced fluid mechanics model, that is, the incompressible Navier-Stokes equations driven by nonlinear colored noise on a two-dimensional unbounded strip domain. This noise is defined on a typical probability space which is not incomplete. The nonlinear function governing the noise intensity is not assumed to be Lipschitz continuous, potentially resulting in non-unique solutions. This type of equation can be regarded as an approximated model of the incompressible Navier-Stokes equations driven by nonlinear white noise, while the long-term behavior of the latter, particularly regarding global attractor theory, remains highly challenging. Our main objective is to investigate the influence of nonlinear colored noise on the long-time behavior in the framework of infinite-dimensional dynamical systems, which is important in understanding the turbulent fluid. We define an appropriate multi-valued random evolutionary semiflow to address the issue caused by the non-uniqueness of solutions, prove the measurability of such a semiflow by examining the specific continuity of the solution operators, while utilizing the idea of energy equation to overcome the difficulties arising from the unboundedness of the domain. The limit behavior, especially the asymptotically autonomous robustness for the equations on two-dimensional bounded domains with smooth boundary, is also investigated.