Averaging of a Three-Dimensional Brinkman-Forchheimer Equation with Singularly Oscillating Forces

Abstract

We consider the uniform attractors of a 3D non-autonomous Brinkman– Forchheimer equation with a singularly oscillating force $$\frac{\partial u}{\partial t}-\gamma\Delta u+au+b|u|u+c|u|^2u+\nabla p=f_0(x,t)+\varepsilon^{-\rho}f_1\Bigg(x,\frac{t}{\varepsilon}\Bigg),$$ for $ρ∈[0,1)$ and $\varepsilon>0,$ and the averaged equation (corresponding to the limiting case $\varepsilon=0$) $$\frac{\partial u}{\partial t}-\gamma\Delta u+au+b|u|u+c|u|^2u+\nabla p=f_0(x,t).$$Given a certain translational compactness assumption for the external forces, we obtain the uniform boundedness of the uniform attractor $\mathcal{A}^{\varepsilon}$ of the first system in $(H^1_0(Ω))^3,$ and prove that when $\varepsilon$ tends to 0, the uniform attractor of the first system $\mathcal{A}^{\varepsilon}$ converges to the attractor $\mathcal{A}^0$ of the second system.