The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations

The long time behavior of the solutions to the two dimensional dissipative quasi-geostrophic equations is studied. We obtain a new positivity lemma which improves a previous version of A. Cordoba and D. Cordoba [10] and [11]. As an application of the new positivity lemma, we obtain the new maximum principle, i.e. the decay of the solution in L-p stop for any p is an element of [2,+ infinity) when f is zero. As a second application of the new positivity lemma, for the sub-critical dissipative case with alpha is an element of (1/2, 1], the existence of the global attractor for the solutions in the space H-s for any s > 2(1 - alpha) is proved for the case when the time independent f is non-zero. Therefore, the global attractor is infinitely smooth if f is. This significantly improves the previous result of Berselli [2] which proves the existence of an attractor in some weak sense. For the case alpha = 1, the global attractor exists in H-s for any s greater than or equal to 0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.