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

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.

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Cited by in corpus (1)

• Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method