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Volume 37, Issue 4
Averaging of a Three-Dimensional Brinkman-Forchheimer Equation with Singularly Oscillating Forces

Xueli Song, Xiaofeng Li, Xi Deng & Biaoming Qiao

DOI: 10.4208/jpde.v37.n4.1

J. Part. Diff. Eq., 37 (2024), pp. 355-376.

Published online: 2024-12

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  • 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. 

  • Keywords

Brinkman-Forchheimer equation, uniform attractor, singularly oscillating external force, uniform boundedness.

  • AMS Subject Headings

35B40, 35B41, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-355, author = {Song , XueliLi , XiaofengDeng , Xi and Qiao , Biaoming}, title = {Averaging of a Three-Dimensional Brinkman-Forchheimer Equation with Singularly Oscillating Forces}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {4}, pages = {355--376}, 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. 

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n4.1}, url = {http://global-sci.org/intro/article_detail/jpde/23686.html} }
TY - JOUR T1 - Averaging of a Three-Dimensional Brinkman-Forchheimer Equation with Singularly Oscillating Forces AU - Song , Xueli AU - Li , Xiaofeng AU - Deng , Xi AU - Qiao , Biaoming JO - Journal of Partial Differential Equations VL - 4 SP - 355 EP - 376 PY - 2024 DA - 2024/12 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n4.1 UR - https://global-sci.org/intro/article_detail/jpde/23686.html KW - Brinkman-Forchheimer equation, uniform attractor, singularly oscillating external force, uniform boundedness. AB -

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. 

Song , XueliLi , XiaofengDeng , Xi and Qiao , Biaoming. (2024). Averaging of a Three-Dimensional Brinkman-Forchheimer Equation with Singularly Oscillating Forces. Journal of Partial Differential Equations. 37 (4). 355-376. doi:10.4208/jpde.v37.n4.1
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