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

$\frac{\partial u}{\partial t}-\gamma \Delta u+a u+b|u| u+c|u|^{2} u+\nabla p=f_{0}(x, t)+\varepsilon^{-\rho} f_{1}\left(x, \frac{t}{\varepsilon}\right),$

for ρ∈[0,1) and ε>0, and the averaged equation (corresponding to the limiting case $\epsilon =0$)

$\frac{\partial u}{\partial t}-\gamma \Delta u+a u+b|u| u+c|u|^{2} u+\nabla p=f_{0}(x, t).$

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 $\left(H_{0}^{1}(\Omega)\right)^{3}$, and prove that when ε tends to 0, the uniform attractor of the first system $\mathcal{A}^{\varepsilon}$ converges to the attractor $\mathcal{A}^{0}$ of the second system.

Keywords

Brinkman-Forchheimer equation / uniform attractor / singularly oscillating external force / uniform boundedness