Bulletin of the
Korean Mathematical Society BKMS

This paper is devoted to studying the long-time behavior for the three-dimensional stochastic globally modified Navier-Stokes equations driven by a linear multiplicative white noise on some unbounded domains $\mathcal{O}.$ By using the Ornstein-Uhlenbeck process, we first transfer the original equation to a random dynamical system, and then prove the existence of pullback attractors as well as the upper semicontinuity of the attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, the asymptotic compactness of the solutions is proved by Ball's idea of energy equations. The periodicity of the attractors is also obtained when the deterministic non-autonomous external terms are periodic in time. Our results extend and generalize some existing results

Keywords: Stochastic globally modified Navier-Stokes equations, pullback random attractors, multiplicative noise, unbounded domains

MSC numbers: Primary 35Q35, 35B40, 35B41, 60H15, 60H30

Supported by: This research was funded by Hanoi Pedagogical University 2 (HPU2) under Grant number HPU2.2022-UT-10.