Stochastic 3D Navier-stokes Equations With Damping: Well-posedness And Dynamics

We concern with a classes of stochastic 3D Navier-Stokes equation with damp-ing in this paper. The damping is from the resistance to the motion of the flow. It describes various physical situations such as drag or friction effects, and some dis-sipative mechanisms. First, the stochastic 3D Navier-Stokes equation with damping driven by a multiplicative noise is considered. Finally, we investigate the class of stochastic 3D Navier-Stokes equation with damping driven by a jump noise. This thesis, constructed by four parts, is organized as follows:In chapter 1, the backgrounds of our investigations and some preliminary in-equalities are given.In chapter 2, by using classical Faedo-Galerkin approximation and compactness method, the existence of martingale solutions for the stochastic 3D Navier-Stokes e-quations with damping is obtained. We conquer some difficulties (such as the estimate of ?0t?D|u·(?)u|2dxds) and prove the existence and uniqueness of strong solution for? > 3 with any ? > 0 and ??1/2 as ? = 3. Meanwhile, a small time large deviation principle for the stochastic 3D Navier-Stokes equations with damping is obtained for? > 3 with any ? > 0 and ??> 1/2as ? = 3.In chapter 3, a stochastic 3D Navier-Stokes equation with damping driven by a multiplicative noise is considered. By using monotonicity method, we conquer some difficulties (such as the estimate of ?0t?D|u·(?)u|2dxds) and prove the existence and uniqueness of the solution for ? > 3 with any ? > 0 and ??1/2as ? = 3. Using the Krylov-Bogoliubov method, we overcome some difficulties (such as the estimate of(?) and show the existence of invariant measures for 3 < ?? 5 with any ? > 0 and ? ? as ?=3. Using asymptotic strong Feller property, the uniqueness of invariant measures is obtained for the degenerate additive noise. By overcoming some difficulties (such as the estimate of ?Dz-2(t)|v|2(?)v|2ds), the existence of a random attractor for the random dynamical systems generated by the solution of stochastic Navier-Stokes equation with damping is proved for ? > 3 with any ? 0 and ? >as ?= 3. The large deviation principle for 3D stochastic Navier-Stokes equations with damping perturbed by multiplicative noise is considered in this chapter 3. By using weak convergence method, we introduce the inequality (?)and prove a large deviation principle for 3D stochastic Navier-Stokes equations with damping for 3 ??<5.In chapter 4, the class of stochastic 3D Navier-Stokes equation with damping driven by a noise of Levy type is considered. We show the function g(s) is smooth and satisfiesBy using the estimateof ?0t?D|u·(?)u|2dxds,we overcome the main difficulty driven by jump noise. The existence and uniqueness of the solution for the problem are proved for ? > 2 with any ? > 0 and ??1/2 as ?= 2. The main result can be applied to various types of SPDE such as stochastic 3D Navier-Stokes equation with damping, stochastic tamed 3D Navier-Stokes equation, stochastic three-dimensional Brinkman-Forchheimer-extended Darcy model. By using the exponential stability of solutions, the existence of a unique invariant measures is proved.