The Research On The Problems Of Attractors For The G-Navier-Stokes Equations

2D g-Navier-Stokes equations is derived from the study of 3D Navier-Stokes equations in thin domain,many good properties of this equations promote the study of 3D Navier-Stokes equations.In this thesis,the problems of attractors for 2D g-Navier-Stokes equations with a nonlinear damping c|u|~?u were investigated.Firstly,2D g-Navier-Stokes equations with a nonlinear damping c|u|~? are investigated on some unbounded domains.By proving the existence of bounded absorbing set for solution semi-group,and then verifying the semi-group satisfies the asymptotic compactness,it is obtained the existence of global attractor.Secondly,the dimensions of the global attractor are estimated.By proving that the mapping V from the bounded closed set M in the separable Hilbert space X to X is Lipschitz continuous and there is a compact semi-norm on X,we estimate the Fractal dimension of the global attractor directly,Moreover,the estimation of the Hausdorff dimension of global attractor was also obtained.Finally,we discussed the exponential attractor of 2D g-Navier-Stokes equations with a nonlinear damping c|u|~?u under periodic boundary condition based on the existence theorem of exponential attractor.After proving the existence of solution and absorbing set,the decomposition technique is used to prove the compact smoothness of the semi-group,and the existence of exponential attractor is obtained.