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arXiv 2203.03031math-phmath.APmath.MP
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大数据的矩传播与相对论Vlasov方程的半经典极限

Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation

Nikolai Leopold, Chiara Saffirio

发表信息：
SIAM J. Math. Anal. 55(3), 1676-1706 (2023)
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AI总结：

该研究推导半相对论Hartree-Fock方程的半经典极限，证明混合态下其以显式Schatten范数速率收敛于带奇异势的相对论Vlasov方程解的Weyl变换，推广了光滑势情形的已有结果，并给出该奇异相对论Vlasov方程适定性的新结论。

摘要：

我们研究了半相对论Hartree-Fock方程的半经典极限，该方程描述平均场区域中费米子系统的时间演化，系统具有相对论色散律，并通过形式为$K(x)=γ\frac{1}{|x|^a}$的奇异势相互作用，其中$a \in \left( \max \left\{ \frac{d}{2} -2 , - 1 \right\}, d-2 \right]$，$d\in\{2,3\}$且$γ\in\mathbb{R}$，约定当$a=0$时$K(x)=γ\log(|x|)$。对于混合态，我们证明了在Schatten范数下以显式速率收敛于带奇异势的相对论Vlasov方程解的Weyl变换，从而推广了[J. Stat. Phys. 172 (2), 398--433 (2018)]中处理光滑势的情况。此外，我们还给出了带奇异相互作用的相对论Vlasov方程适定性理论的新结果。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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