Averaging of a multi-layer quasi-geostrophic equations  withoscillating external forces

T. Tachim Medjo

doi:10.3934/cpaa.2014.13.1119

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2014, Volume 13, Issue 3: 1119-1140. Doi: 10.3934/cpaa.2014.13.1119

This issue Previous Article Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients Next Article Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

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

T. Tachim Medjo^1,

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

Received: March 31, 2013

Revised: August 31, 2013

Published: April 2015

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Abstract

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].

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Mathematics Subject Classification: Primary: 35Q30,35Q35; Secondary: 35Q72.

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