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The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. (English) Zbl 1088.37049
The article deals with the 2D quasi-geostrophic equations with dissipation, i.e., \(\theta_t + u\cdot \nabla \theta + \kappa (-\Delta)^\alpha \theta = f\) in \((0,T)\times \Omega\), where \(u=(-\frac{\partial \psi}{\partial x_2}, \frac{\partial \psi}{\partial x_1})\), \((-\Delta)^\frac {1}{2} \psi = \theta\) and either \(\Omega =\mathbb{R}^2\) or \(\Omega\) a rectangle with space periodic boundary conditions.
The author improves the positivity lemma shown by A. Córdoba and D. Córdoba [Proc. Natl. Acad. Sci. USA 100, No. 26, 15316–15317 (2003; Zbl 1111.26010)] \[ \int_\Omega | \theta| ^{p-2}\theta (-\Delta)^{s/2} \theta dx \geq \frac{C}{p} \int_\Omega | (-\Delta)^{s/4} \theta^{p/2}| ^2 dx \] in two aspects. The constant \(C=2\) (while in the above cited paper \(C=1\)) and it holds for any \(p\geq 2\) (while in the above cited paper \(p=2^n\), \(n\) natural number).
Thus, a solution in \(L^p(\Omega)\) decays to zero. Further, due to the positivity lemma, the author is able to prove the existence of the global attractor in the space \(H^s(\Omega)\) for all \(s>2(1-\alpha)\).

MSC:
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
35Q35 PDEs in connection with fluid mechanics
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
86A05 Hydrology, hydrography, oceanography
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