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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.21938 |
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| _version_ | 1866909976274927616 |
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| author | Hu, Zheng-Nan Jang, Jin Woo Yao, Zheng-An Zhou, Yu-Long |
| author_facet | Hu, Zheng-Nan Jang, Jin Woo Yao, Zheng-An Zhou, Yu-Long |
| contents | The inverse power potential $U(r)=r^{-1/s}, 0<s<1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(θ)$ with an angular singularity as $θ\to 0$. Jang-Kepka-Nota-Velázquez (2023) proved the limit $B^{s}\to \frac14|v-v_*|$ as $s\to 0$, as well as weak convergence of solutions based on this kernel convergence. In this work we establish the following sharp quantitative estimate: $$ |b_s(θ)-\tfrac14| \le C\, s\,θ^{-2-2s}. $$ In particular, this sharp estimate yields the optimal $O(s)$ convergence rate for solutions of the homogeneous Boltzmann equation with large initial data in suitable Sobolev spaces; i.e., for any $t\in[0,T]$, we have $$f^s(t)=f^0(t)+O(s),$$ quantified by the $L^1_k$ norm for $k\ge 2.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation Hu, Zheng-Nan Jang, Jin Woo Yao, Zheng-An Zhou, Yu-Long Analysis of PDEs The inverse power potential $U(r)=r^{-1/s}, 0<s<1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(θ)$ with an angular singularity as $θ\to 0$. Jang-Kepka-Nota-Velázquez (2023) proved the limit $B^{s}\to \frac14|v-v_*|$ as $s\to 0$, as well as weak convergence of solutions based on this kernel convergence. In this work we establish the following sharp quantitative estimate: $$ |b_s(θ)-\tfrac14| \le C\, s\,θ^{-2-2s}. $$ In particular, this sharp estimate yields the optimal $O(s)$ convergence rate for solutions of the homogeneous Boltzmann equation with large initial data in suitable Sobolev spaces; i.e., for any $t\in[0,T]$, we have $$f^s(t)=f^0(t)+O(s),$$ quantified by the $L^1_k$ norm for $k\ge 2.$ |
| title | Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.21938 |