, Volume 255, Issue 1, pp 161-181
Date: 11 Jan 2005

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

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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 L p for any p ∈ [2,+∞) when f is zero. As a second application of the new positivity lemma, for the sub-critical dissipative case with the existence of the global attractor for the solutions in the space H s 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 H s for any s≥0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.

Communicated by P. Constantin
Acknowledgement The author thanks Prof. P. Constantin for encouragement and kind help for his research on the subject of 2D QG equations, Prof. J. Wu for useful conversation and Prof. A. Cordorba for providing preprints. This work was started while the author visited IPAM at UCLA with an IPAM fellowship. The hospitality and support of IPAM is gratefully acknowledged. This work is partially supported by the Oklahoma State University new faculty start-up fund and the Dean’s Incentive Grant.