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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 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 86A05 Hydrology, hydrography, oceanography
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