Pullback Attractor For Stochastic Navier-Stokes Equation Driven By Fractional Brownian Motion

On a bounded domain G∈R2, it is consider the long time dynamic behavior of the following non-autonomous Stochastic Navier-Stokes equation Where stochastic term Bh is the infinite dimensional fractional Brownian motion with Hurst parameter h∈(0,1/2),f is non-autonomous deterministic terms with time-independent, v>0is the viscosity of the fluids, and P is the pressureIn the1990s, F. FLandoli has studied the long dynamic behavior of Navier-Stokes equation driven by standard Brownian motion. Then, fractional Brownian motion is widely present phenomenon in nature. The existence of attractor for Navier-Stokes equation driven by fractional Brownian motion has not been researched. Under suitable condition in this paper, we use the uniform estimates method to prove the existence of the random attractor and solution for the stochastic dynamical system generated by above equation. In order to prove this result, we need the semigroup S(t) with respect to Wiener integral, which be generated by the equation, is continuous predictable Gaussian processes in the sense of mean square.