LIMITING DYNAMICS FOR STOCHASTIC NAVIER-STOKES EQUATIONS ON EXPANDING UNBOUNDED DOMAINS

We study the limiting dynamics of stochastic non-autonomous Navier-Stokes equations defined on a sequence of expanding domains, where the largest is an unbounded Poincare<acute accent> domain. We prove the upper semi-continuity of the null-expansion of the corresponding random attractor when the bounded domain is expanded to the unbounded domain. To do this, we expand each random dynamical system (co cycle) and then prove the expanding co cycle converges to the co cycle on the unbounded domain. By generalizing the famous energy equation method, we prove that the sequence of expanding co cycles is weakly equi-continuous and strongly equi-asymptotically compact, which lead to the upper semi-continuity of attractors.