Asymptotic behaviour of solutions to the 2D stochastic Navier-Stokes equations in unbounded domains - New developments

This paper is concerned with the large time behaviour of solutions to the 2D stochastic Navier-Stokes equation on some unbounded domains. A notion of asymptotic compactness of a Random Dynamical System on a Polish space, introduced by authors in [11], is recalled. We present some results from that paper as well as announce some results from [12]. Firstly, we show that an asymptotically compact RDS on a separable Banach space has a non-empty compact Q-limit set. Secondly, we show that the RDS generated by a stochastic NSEs on a 2D domain satisfying the Poincare inequality is asymptotically compact on the space H of divergence free vector fields on D with finite L-2 norm. Thirdly, we show that this is also true for the space V of divergence free vector fields on D with finite H-1,H-2 norm. The first two results have been proved in [11], the third comes from the work in preparation [12]. This research is motivated by papers [35] by Rosa and [31] by Ju, where similar questions for dynamical systems generated by 2D deterministic NSEs were studied. One consequence of our main results is the existence of a Feller invariant measures (however the uniqueness of invariant measures remains an open problem). The case of stochastic NSEs in bounded domains have been studied earlier in [4], [8] and [38].