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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:
 [1] T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171 [2] Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133 [3] May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179 [4] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [5] Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197 [6] T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265 [7] Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023 [8] T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415 [9] Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525 [10] Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277 [11] Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293 [12] Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152 [13] Djano Kandaswamy, Thierry Blu, Dimitri Van De Ville. Analytic sensing for multi-layer spherical models with application to EEG source imaging. Inverse Problems & Imaging, 2013, 7 (4) : 1251-1270. doi: 10.3934/ipi.2013.7.1251 [14] Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669 [15] François Bouchut, Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 739-758. doi: 10.3934/dcdsb.2010.13.739 [16] Jinsen Zhuang, Yan Zhou, Yonghui Xia. Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020279 [17] Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241 [18] V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27 [19] Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669 [20] T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665

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