Averaging of a 3D primitive equations with oscillating external forces

Abstract

In this article, we consider a non-autonomous three-dimensional primitive equations of the ocean with a singularly oscillating external force g ^ε = g ₀(t) + ε^−ρ g ₁(t/ε) depending on a small parameter ε > 0 and ρ ∈ [0, 1) together with the averaged system with the external force g ₀(t), formally corresponding to the case ε = 0. Under suitable assumptions on the external force, we prove as in [V.V. Chepyzhov, V. Pata, and M.M.I. Vishik, Averaging of 2D Navier–Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), pp. 351–370] the boundness of the uniform global attractor 𝒜^ε as well as the convergence of the attractors 𝒜^ε of the singular systems to the attractor 𝒜⁰ of the averaged system as ε → 0⁺. When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by Kε^(1−ρ). Let us note that the main difference between this work and that of Chepyzhov et al. (2009) is that the non-linearity involved in the three-dimensional primitive equation is stronger than the one in the two-dimensional Navier–Stokes equations considered in Chepyzhov et al. (2009), which makes the analysis of the problem studied in this article more involved.

Keywords:

• averaging
• 3D primitive equations
• global attractor
• oscillating force

AMS Subject Classifications::

• 35Q30
• 35Q35
• 35Q72

Acknowledgements

The author would like to thank the anonymous referees whose comments help to improve the contain of this article.