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

TL;DR

This paper establishes the semiclassical limit from a semi-relativistic Hartree-Fock model to the relativistic Vlasov equation with singular potentials, providing convergence rates and new well-posedness results for such equations.

Contribution

It generalizes previous smooth-potential results to singular potentials and offers explicit convergence rates and well-posedness insights for the relativistic Vlasov equations.

Findings

Proves convergence in Schatten norms with explicit rates.

Extends analysis to singular interaction potentials.

Provides new well-posedness results for relativistic Vlasov equations.

Abstract

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 $K(x)=\gamma\frac{1}{|x|^a}$, $a \in \left( \max \left\{ \frac{d}{2} -2 , - 1 \right\}, d-2 \right]$, $d\in\{2,3\}$ and $\gamma\in\mathbb{R}$, with the convention $K(x)=\gamma\log(|x|)$ 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.