Long Time Behavior For A Modified Critical Dissipative Surface Quasi-Geostrophic Equation

In this paper,we consider the long time behavior for the following modified critical dissipative quasi-geostrophic equation where T² is a bounded periodic torus in R²,???1,2? are real parameters,? are constants,?=?-??^1/2.Firstly,we proof the existence of solutions for the above equation by classical Faedo-Galerkin approximation method;Secondly,using De Giorgi iteration method,we show the estimates of the viscous solution with re-spect to the L^? norm;Then the continuity of viscous solution is obtained by combining Littlewood-Paley decomposition with the preceding estimates of vis-cous solution.Thus we can obtain the existence of the attractor for the equation under the framework of evolutionary system.