Long Time Behavior Of Solutions For Critical Quasi-geostrophic Equation

In this paper,we investigate the global well-posedness,the existence of global at-tractors and its finite dimensions for the critical quasi-geostrophic equation.This paper is divided into four parts.In chapter 1,we firstly introduce the preliminary and some basic theory,and then give the main results of this paper.In chapter 2,we firstly obtain the local well-posedness of the solutions by using the continuity method.We then apply the nonlinear lower bound on the fractional Laplacian and the property of Only Small Shocks to prove the global well-posedness of the solutions.In chapter 3,the existence of the global attractor is considered.Firstly,we get the existence of a compact absorbing set by applying the regularity lifting argument and proving the uniform boundedness of the solutions in H1(R2).Secondly,decomposing the whole space into a bounded ball and its complement with the help of a suitable cut-off function,and then we prove the asymptotic compactness of the semigroup and obtain the existence of the global attractor.In chapter 4,we obtain the finite Hausdorff and fractal dimensions of the global attractor via the fractional Lieb-thirring inequality which is introduced in[11].