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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.09270 |
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| _version_ | 1866915287185489920 |
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| author | Pan, Xinghong Wang, Xue Ping Zhu, Lu |
| author_facet | Pan, Xinghong Wang, Xue Ping Zhu, Lu |
| contents | We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in $\mathbb R^n$, $n \ge 4$, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted $L^2$-spaces when $ n\ge 5$ is odd. For potentials decaying like $O(|x|^{-ρ})$ for some $ρ> n-1$, we obtain, for all dimensions $n \ge 4$, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor $(4πt)^{-\frac n 2}$ corresponding to the decay for the heat equation. These results complete those obtained in [16, 22] for dimensions $n=1$ and $3$. The same questions for $n=2$ are still open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Kramers-Fokker-Planck equation with a decaying potential in $\mathbb R^n$, $n \ge 4$ Pan, Xinghong Wang, Xue Ping Zhu, Lu Analysis of PDEs We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in $\mathbb R^n$, $n \ge 4$, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted $L^2$-spaces when $ n\ge 5$ is odd. For potentials decaying like $O(|x|^{-ρ})$ for some $ρ> n-1$, we obtain, for all dimensions $n \ge 4$, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor $(4πt)^{-\frac n 2}$ corresponding to the decay for the heat equation. These results complete those obtained in [16, 22] for dimensions $n=1$ and $3$. The same questions for $n=2$ are still open. |
| title | The Kramers-Fokker-Planck equation with a decaying potential in $\mathbb R^n$, $n \ge 4$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.09270 |