The Maximum Principle and the Global Attractor for the Dissipative 2D Quasi-Geostrophic Equations

TLDR: In this paper, the authors studied the long time behavior of the solutions to the two dimensional dissipative quasi-geostrophic equations and obtained a new positivity lemma which improved a previous version of A. Cordoba and D. Duygulu.

Abstract: 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 Lp for anyp ∈ [2,+∞) when f is zero. As a second application of the new positivity lemma, for the sub-critical dissipative case with Open image in new window the existence of the global attractor for the solutions in the space Hs for any s>2(1−α) 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 α=1, the global attractor exists in Hs for any s≥0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.