Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917413389336576 |
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| author | Höfer, Richard M. Hübner-Rosenau, Eleni |
| author_facet | Höfer, Richard M. Hübner-Rosenau, Eleni |
| contents | We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^α=N^{-α/3}$ and the viscosity $\varepsilon^γ=N^{-γ/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($α+γ>3$) and critical ($α+γ=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $α>2$ so that the particles do not overlap with overwhelming probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14792 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit Höfer, Richard M. Hübner-Rosenau, Eleni Analysis of PDEs We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^α=N^{-α/3}$ and the viscosity $\varepsilon^γ=N^{-γ/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($α+γ>3$) and critical ($α+γ=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $α>2$ so that the particles do not overlap with overwhelming probability. |
| title | Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.14792 |