On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910294319562752 |
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| author | Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara |
| author_facet | Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara |
| contents | Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the convergence in the $L^2$ norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence proportional to $\hbar$. This improves the $\hbar^{3/4-\varepsilon}$ rate of convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree$\unicode{x2013}$Fock equation for mixed states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11485 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation Chong, Jacky J. Lafleche, Laurent Saffirio, Chiara Analysis of PDEs Mathematical Physics Quantum Physics 81Q20, 35Q55, 35Q83 (Primary) 82C10, 82C05 (Secondary) Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the convergence in the $L^2$ norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence proportional to $\hbar$. This improves the $\hbar^{3/4-\varepsilon}$ rate of convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree$\unicode{x2013}$Fock equation for mixed states. |
| title | On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation |
| topic | Analysis of PDEs Mathematical Physics Quantum Physics 81Q20, 35Q55, 35Q83 (Primary) 82C10, 82C05 (Secondary) |
| url | https://arxiv.org/abs/2203.11485 |