Asymptotic Dynamics Of Globally Modified3-D Stochastic Navier-Stokes Equations With Delay

The Navier-Stokes equations describes the motion law of viscous fluid such as water, oil, and air, which has important significance in Fluid Dynamics. Now, the theories on the two dimensions Navier-Stokes equations have been well developed. However, the theories on the three dimensions Navier-Stokes equations need to be further investigated. In this paper, we mainly discuss the asymptotic dynamics of globally modified3-D Navier-Stokes equations with delay and stochastic perturbations. We study the existence and periodicity of random attractors for the cases with finite delay and infinite delay, respectively. The paper is organized as follows:In Chapter1. we first introduce the research background of Navier-Stokes equations and research situation of random dynamical system. Then some preliminary concepts and results related to random dynamical system are in-troduced.In Chapter2, we study the globally modified3-D stochastic Navier-Stokes equations with finite delay. Firstly, the original equations is transformed into an abstract form by introducing some notations and operators. And through the Ornstein-Uhlenbeck process, the original equations is transform into the equations with random coefficients in order to facilitate the following discus-sion on random dynamical system. Then, we study the global well-posedness of solutions. Moreover, we establish random dynamical system (continuous cocycle) on the space CH. Finaly, by the uniform estimates established for solutions, we prove the existence, uniqueness and periodicity of pullback at- tractors.In Chapter3, we study the globally modified3-D stochastic Navier-Stokes equations with infinite delay. Since the delay is infinite, the definition space Cy(H) of random dynamical system is different from that in Chapter2. More-over, the method is also different when discussing the asymptotic compact property of random dynamical system.