Surface Quasi-geostrophic(SQG) 方程是大气与海洋研究中的重要数学模型，研究其解的长时间行为有助于理解甚至预测混沌系统的状态。临界SQG方程的耗散项与非线性项含有导数的阶一致，且都具有非局部性，因而吸收集的存在性以及解半群的渐近紧性的证明将遇到新的困难。我们拟分别利用Constantin-Vicol非线性极大值原理和频段层次的Bernstein不等式建立全空间中SQG方程解在Hs和Lp中的光滑效应，应用分数拉普拉斯算子的莱布尼兹法则和乘积公式证明解的尾端估计，从而得到SQG方程在Hs和Lp中整体吸引子的存在性。我们将再采用类似的估计技巧研究QG方程不同解的差，证明解半群满足拟稳定性质和锥挤压性质，以此说明吸引子的分形维数有限以及指数吸引子的存在性。

Surface quasi-geostrophic (SQG) equation is an important model in the study of atmosphere and ocean. The research on the long time dynamics of SQG equation will help us to understand and even predict the states of chaotic system. The dissipative term and nonlinear term of critical SQG equation are nonlocal and comparable to each other. Thus, it’s not easy to prove the existence of an absorbing set and asymptotic compactness of solution semigroup. We shall establish the smoothing effect of solution for critical SQG equation on R2 in Hs by Constantin-Vicol nonlinear maximum principle, in Lp by Bernstein inequality with localized frequencies. Then, using fractional Leibniz rule and product formula, we shall obtain the tail estimates of solutions. Combining these together, we will get the existence of global attractor for critical SQG equation in Hs and Lp. Finally, we shall study the difference of two solutions of critical SQG equation by similar techniques, and show the quasi-stable property and squeezing property of solution semigroup, then get the finite dimensionality of global attractor and the existence of exponential attractor.