Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits
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| Format: | Preprint |
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2025
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| _version_ | 1866909457067278336 |
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| author | He, Ling-Bing Jiang, Ning Wu, Yulong |
| author_facet | He, Ling-Bing Jiang, Ning Wu, Yulong |
| contents | This paper investigates the Knudsen layer equation in half-space, arising from the hydrodynamic limit of the Boltzmann equation to fluid dynamics. We consider the Maxwell reflection boundary condition with accommodation coefficient $0<α<1$. We restrict our attention to hard sphere collisions with angular cutoff, proving the existence, uniqueness, and asymptotic behavior of the solution in $L^{\infty}_{x,v}$. Additionally, we demonstrate the application of our theorem to the hydrodynamic limit through a specific example. In this expample, we derive the boundary conditions of the fluid equations using our theorem and the symmetric properties of the Knudsen layer equation for $α\in(0,1]$ and $α=O(1)$. These derivations differs significantly from the cases of specular and almost specular reflection. This explicitly characterizes the {\em vanishing sources set} defined in \cite{jiang2024knudsenboundarylayerequations} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_08733 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits He, Ling-Bing Jiang, Ning Wu, Yulong Analysis of PDEs 35Q20, 76P05, 35F30, 35A01, 35A02 This paper investigates the Knudsen layer equation in half-space, arising from the hydrodynamic limit of the Boltzmann equation to fluid dynamics. We consider the Maxwell reflection boundary condition with accommodation coefficient $0<α<1$. We restrict our attention to hard sphere collisions with angular cutoff, proving the existence, uniqueness, and asymptotic behavior of the solution in $L^{\infty}_{x,v}$. Additionally, we demonstrate the application of our theorem to the hydrodynamic limit through a specific example. In this expample, we derive the boundary conditions of the fluid equations using our theorem and the symmetric properties of the Knudsen layer equation for $α\in(0,1]$ and $α=O(1)$. These derivations differs significantly from the cases of specular and almost specular reflection. This explicitly characterizes the {\em vanishing sources set} defined in \cite{jiang2024knudsenboundarylayerequations} |
| title | Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits |
| topic | Analysis of PDEs 35Q20, 76P05, 35F30, 35A01, 35A02 |
| url | https://arxiv.org/abs/2501.08733 |