Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation
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arXiv
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| Natura: | Preprint |
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2022
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| author | Leopold, Nikolai Saffirio, Chiara |
| author_facet | Leopold, Nikolai Saffirio, Chiara |
| contents | We investigate the semiclassical limit from the semi-relativistic Hartree-Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form $K(x)=γ\frac{1}{|x|^a}$, $a \in \left( \max \left\{ \frac{d}{2} -2 , - 1 \right\}, d-2 \right]$, $d\in\{2,3\}$ and $γ\in\mathbb{R}$, with the convention $K(x)=γ\log(|x|)$ if $a=0$. For mixed states, we show convergence in Schatten norms with explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [J. Stat. Phys. 172 (2), 398--433 (2018)] where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_03031 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation Leopold, Nikolai Saffirio, Chiara Mathematical Physics Analysis of PDEs 35Q41, 82C10, 35Q55, 35Q83, 70H40, 81Q20 We investigate the semiclassical limit from the semi-relativistic Hartree-Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form $K(x)=γ\frac{1}{|x|^a}$, $a \in \left( \max \left\{ \frac{d}{2} -2 , - 1 \right\}, d-2 \right]$, $d\in\{2,3\}$ and $γ\in\mathbb{R}$, with the convention $K(x)=γ\log(|x|)$ if $a=0$. For mixed states, we show convergence in Schatten norms with explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [J. Stat. Phys. 172 (2), 398--433 (2018)] where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions. |
| title | Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation |
| topic | Mathematical Physics Analysis of PDEs 35Q41, 82C10, 35Q55, 35Q83, 70H40, 81Q20 |
| url | https://arxiv.org/abs/2203.03031 |