Averaging Principle For Two Dimensional Stochastic Navier-Stokes Equations With Multiplicative Noise

The theory of averaging principle has widely applied in materials science,chem-istry,fluid dynamics,biology,ecology,climate dynamics,ect.So,the averaging prin-ciple of stochastic partial differential equations has attracted the attention and research of stochastic analysis experts and scholars at home and abroad.However,almost all re-search results are mostly concentrated in the linear case and the results on the nonlinear systems are few.This paper studies averaging principle of two dimensional stochastic Navier-Stokes equations that driven by multiplicative noise,that is,the averaging principle has two time scales,where the slow component is two dimensional stochastic Navier-Stokes equations,and the fast component is a stochastic reaction-diffusion equation.It is difficult to study it because Navier-Stokes equation is local nonlinear.Under the condition of releasing the the initial value regularity,we show that the slow component strong converges to the solution of the corresponding averaged equation by the tech-nology of stopping time.It is based on the Khasminskii discretization argument and it is a new result.The main result can be proved by the following two steps.Step 1,we will use stopping time techniques and control the |X_t~?-(?)_t~?| and |(?)_t~?-(?)_t| respectively.Step2,after the stopping time term can be estimated by the priori estimates of the solution.