Non-autonomous 3D Brinkman-Forchheimer equation with singularly oscillating external force and its uniform attractor

Abstract

For $\rho\in [0,1)$ and $\varepsilon>0$, the non-autonomous 3D Brinkman-Forchheimer equation with singularly oscillating external force \begin{align*} &u_t-\nu\Delta u+au+b|u|u+c|u|^\beta u+\nabla p=f_0(x,t)+\varepsilon^{-\rho}f_1(x,\frac{t}{\varepsilon}),\\ &\mathrm{div}u=0\end{align*} are considered, together with the averaged equation \begin{align*}&u_t-\nu\Delta u+au+b|u|u+c|u|^\beta u+\nabla p=f_0(x,t),\\ &\mathrm{div}u=0\end{align*} formally corresponding to the limiting case $\varepsilon=0$. First, within the restriction $\rho<1$ and under suitable translation-compactness assumptions on the external forces, the uniform (w.r.t.$\varepsilon$) boundedness of the related uniform attractors $\mathcal{A}^\varepsilon$ is established when $1<\beta\leq 4/3$. This fact is not at all intuitive, since in principle the blow up of the oscillation amplitude might overcome the averaging effect due to the term $\frac{t}{\varepsilon}$ in $f_1$. Next, the convergence of the attractor $\mathcal{A}^\varepsilon$ of the first equation to the attractor $\mathcal{A}^0$ of the second one as $\varepsilon\rightarrow 0^+$ is established.

Keywords

• 3d brinkman-forchheimer equation
• singularly oscillating external force
• uniform attractor
• translation bounded function