Unified quantitative analysis of the Stokes equations in dilute perforated domains via layer potentials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929605273714688 |
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| author | Jing, Wenjia Lu, Yong Prange, Christophe |
| author_facet | Jing, Wenjia Lu, Yong Prange, Christophe |
| contents | We develop a unified method to obtain the quantitative homogenization of Stokes systems in periodically perforated domains with no-slip boundary conditions on the perforating holes. The main novelty of our paper is a quantitative analysis of the asymptotic behavior of the two-scale cell correctors via periodic Stokes layer potentials. The two-scale cell correctors were introduced and analyzed qualitatively by Allaire in the early 90's. Thanks to our layer potential approach, we also provide a novel explanation of the conductivity matrix in Darcy's model, of the Brinkman term in Brinkman's model, and explain the special behavior for $d=2$. Finally, we also prove quantitative homogenization error estimates in various regimes of ratios between the size of the perforating holes and the typical distance between holes. In particular we handle a subtle issue in the dilute Darcy regime related to the non-vanishing of the Darcy velocity on the boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_16960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unified quantitative analysis of the Stokes equations in dilute perforated domains via layer potentials Jing, Wenjia Lu, Yong Prange, Christophe Analysis of PDEs 35B27, 35B40, 35J08, 35Q35, 76D07 We develop a unified method to obtain the quantitative homogenization of Stokes systems in periodically perforated domains with no-slip boundary conditions on the perforating holes. The main novelty of our paper is a quantitative analysis of the asymptotic behavior of the two-scale cell correctors via periodic Stokes layer potentials. The two-scale cell correctors were introduced and analyzed qualitatively by Allaire in the early 90's. Thanks to our layer potential approach, we also provide a novel explanation of the conductivity matrix in Darcy's model, of the Brinkman term in Brinkman's model, and explain the special behavior for $d=2$. Finally, we also prove quantitative homogenization error estimates in various regimes of ratios between the size of the perforating holes and the typical distance between holes. In particular we handle a subtle issue in the dilute Darcy regime related to the non-vanishing of the Darcy velocity on the boundary. |
| title | Unified quantitative analysis of the Stokes equations in dilute perforated domains via layer potentials |
| topic | Analysis of PDEs 35B27, 35B40, 35J08, 35Q35, 76D07 |
| url | https://arxiv.org/abs/2409.16960 |