On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917748738621440 |
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| author | Cao, Chuqi He, Ling-Bing Ji, Jie |
| author_facet | Cao, Chuqi He, Ling-Bing Ji, Jie |
| contents | We provide the first and rigorous confirmations of the hypotheses by Ludwig Boltzmann in his seminal paper \cite{Boltzmann} within the context of the Landau equation in the presence of a harmonic potential. We prove that (i) Each {\it entropy-invariant solution} can be identified as a {\it time-periodic Maxwell-Boltzmann distribution}. Moreover, these distributions can be characterized by thirteen conservation laws, which sheds light on the global dynamics. (ii) Each {\it time-periodic Maxwell-Boltzmann distribution} is nonlinearly stable, including neutral asymptotic stability and Lyapunov stability. Furthermore, the convergence rate is entirely reliant on the thirteen conservation laws and is optimal when compared to the linear scenario. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions Cao, Chuqi He, Ling-Bing Ji, Jie Analysis of PDEs 35Q20, 82B40, We provide the first and rigorous confirmations of the hypotheses by Ludwig Boltzmann in his seminal paper \cite{Boltzmann} within the context of the Landau equation in the presence of a harmonic potential. We prove that (i) Each {\it entropy-invariant solution} can be identified as a {\it time-periodic Maxwell-Boltzmann distribution}. Moreover, these distributions can be characterized by thirteen conservation laws, which sheds light on the global dynamics. (ii) Each {\it time-periodic Maxwell-Boltzmann distribution} is nonlinearly stable, including neutral asymptotic stability and Lyapunov stability. Furthermore, the convergence rate is entirely reliant on the thirteen conservation laws and is optimal when compared to the linear scenario. |
| title | On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions |
| topic | Analysis of PDEs 35Q20, 82B40, |
| url | https://arxiv.org/abs/2407.07659 |