Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912809993895936 |
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| author | Bihan, Corentin Le |
| author_facet | Bihan, Corentin Le |
| contents | We provide a rigorous justification of the linearized Boltzmann- and Landau equations from interacting particle systems with long-range interaction. The result shows that the fluctuations of Hamiltonian $N$- particle systems governed by truncated power law potentials of the form $\mathcal{U}(r)\sim |r/\varepsilon|^{-s}$ (near $r \approx 0$) converge to solutions of kinetic equations in appropriate scaling limits $\varepsilon \rightarrow 0$ and $N\rightarrow \infty$. We prove that for $s\in [0,1)$, the limiting system approaches the uncutoff linearized Boltzmann equation or the linearized Landau equation, depending on the scaling limit. The Coulomb singularity $s=1$ appears as a threshold value. Kinetic scaling limits with $s\in (0,1]$ universally converge to the linearized Landau equation, and we prove the onset of the Coulomb logarithm for $s=1$. To the best of our knowledge, this is the first result on the derivation of kinetic equations from interacting particle systems with long-range power-law interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law Bihan, Corentin Le Analysis of PDEs We provide a rigorous justification of the linearized Boltzmann- and Landau equations from interacting particle systems with long-range interaction. The result shows that the fluctuations of Hamiltonian $N$- particle systems governed by truncated power law potentials of the form $\mathcal{U}(r)\sim |r/\varepsilon|^{-s}$ (near $r \approx 0$) converge to solutions of kinetic equations in appropriate scaling limits $\varepsilon \rightarrow 0$ and $N\rightarrow \infty$. We prove that for $s\in [0,1)$, the limiting system approaches the uncutoff linearized Boltzmann equation or the linearized Landau equation, depending on the scaling limit. The Coulomb singularity $s=1$ appears as a threshold value. Kinetic scaling limits with $s\in (0,1]$ universally converge to the linearized Landau equation, and we prove the onset of the Coulomb logarithm for $s=1$. To the best of our knowledge, this is the first result on the derivation of kinetic equations from interacting particle systems with long-range power-law interaction. |
| title | Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.03597 |