The Existence Of The Invariant Measures For2D Stochastic Navier-Stokes-Burgers Equation

We consider the following2-dimensional Navier-Stokes-Burgers equation for an in-compressible fuid in a bounded domain D with smooth boundary D du=(△u+1/2▽u2+(u·▽u)dt+dW(t) Where W is a Gaussian form space-time random feld, which is white noise in time, and as general as possible in the space variable. In this paper, Win the case of standard Brownian motion or fractal Brownian motion. We prove the existence of invariant measure for the equeation by using the Krylov-Bogoliubov theorem. We first prove the existence and uniqueness for the solution of2-dimensional Navier-Stokes-Burgers equation driven by the infinite dimensional standard Brownian motion and the property of Feller for transfer semigroup. As the embedding H2θ(D)→L2(D) is compactness,{L(uλ(0))}λ>0is tight, hence the invariant measure is exist. Then prove the existence and uniqueness for the solution of2-dimensional Navier-Stokes-Burgers equation driven by the infinite dimensional fractal Brownian motion. As the embedding H1(D)→L2(D)is compactness,{L(uλ(0))}λ≥0is tight, hence the invariant measure is exist.