Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications
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Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients
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
T. Tachim Medjo 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].
Keywords: global attractor, Averaging, multi-layer quasi-geostrophic, oscillating force.
Mathematics Subject Classification: Primary: 35Q30,35Q35; Secondary: 35Q7.
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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T. Tachim Medjo