Invariant Measures And Trajectory Statistical Solution For The Navier-Stokes Equations

This master thesis studies the invariant measures and trajectory statistical solution for Navier-Stokes equations.The first section ap-plies the approach of enstrophy equations to prove the existence of the pullback attractor for the non-autonomous globally modified Navier-Stokes equations in the norm of H~1,then the paper establishes that the associated process generated by solution operators possesses a family of invariant Borel probability measures on the pullback attractor,and the support of the invariant measures is contained in the pullback attractor.The second section constructs the trajectory statistical solution for the3D incompressible Navier-Stokes equations via the natural translation semigroup and trajectory attractor.In our construction,the trajec-tory statistical solution is an invariant space-time probability measure which is carried by the trajectory attractor of the natural translation semigroup defined on the trajectory space,and the trajectory statis-tical solution possesses the invariant property under the acting of the translation semigroup.