Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866918284494897152 |
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| author | Chen, I-Kun Hsia, Chun-Hsiung Kawagoe, Daisuke Su, Jhe-Kuan |
| author_facet | Chen, I-Kun Hsia, Chun-Hsiung Kawagoe, Daisuke Su, Jhe-Kuan |
| contents | We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in $W^{1,p}$ spaces for $1 \leq p<2$ for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$ provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for $p = 2$, and a small ball for $p = 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12387 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation Chen, I-Kun Hsia, Chun-Hsiung Kawagoe, Daisuke Su, Jhe-Kuan Analysis of PDEs 35Q20, 35B65 We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in $W^{1,p}$ spaces for $1 \leq p<2$ for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$ provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for $p = 2$, and a small ball for $p = 3$. |
| title | Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation |
| topic | Analysis of PDEs 35Q20, 35B65 |
| url | https://arxiv.org/abs/2311.12387 |