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May  2014, 13(3): 1119-1140. doi: 10.3934/cpaa.2014.13.1119

## Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces

 1 Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received  April 2013 Revised  September 2013 Published  December 2013

In this article, we consider a non-autonomous multi-layer quasi-geostrophic equations of the ocean with a singularly oscillating external force $g^{\epsilon}= g_0(t) + \epsilon^{-\rho} g_1(t/\epsilon)$ depending on a small parameter $\epsilon > 0$ and $\rho \in [0, 1)$ together with the averaged system with the external force $g_0(t),$ formally corresponding to the case $\epsilon = 0.$ Under suitable assumptions on the external force, we prove as in [10] the boundness of the uniform global attractor $\mathcal{A}^{\epsilon}$ as well as the upper semi-continuity of the attractors $\mathcal{A}^{\epsilon}$ of the singular systems to the attractor $\mathcal{A}^0$ of the averaged system as $\epsilon \rightarrow 0^+.$ When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by $K \epsilon^{(1 -\rho)}.$ Let us mention that the non-homogenous boundary conditions (and the non-local constraint) present in the multi-layer quasi-geostrophic model makes the estimates more complicated, [3]. These difficulties are overcome using the new formulation presented in [25].
Citation: T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119
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##### References:
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