Vanishing angular singularity limit to the hard-sphere Boltzmann equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917289237938176 |
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| author | Jang, Jin Woo Kepka, Bernhard Nota, Alessia Velázquez, Juan J. L. |
| author_facet | Jang, Jin Woo Kepka, Bernhard Nota, Alessia Velázquez, Juan J. L. |
| contents | In this note we study Boltzmann's collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $ d=3 $. We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near $θ\simeq 0$ in the limit $ s\to \infty $. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14075 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Vanishing angular singularity limit to the hard-sphere Boltzmann equation Jang, Jin Woo Kepka, Bernhard Nota, Alessia Velázquez, Juan J. L. Analysis of PDEs Mathematical Physics 35Q20 (Primary) 35R11, 76P05, 82C40, 26A33 (Secondary) In this note we study Boltzmann's collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $ d=3 $. We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near $θ\simeq 0$ in the limit $ s\to \infty $. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions. |
| title | Vanishing angular singularity limit to the hard-sphere Boltzmann equation |
| topic | Analysis of PDEs Mathematical Physics 35Q20 (Primary) 35R11, 76P05, 82C40, 26A33 (Secondary) |
| url | https://arxiv.org/abs/2209.14075 |