PROPAGATION OF MOMENTS FOR LARGE DATA AND SEMICLASSICAL LIMIT TO THE RELATIVISTIC VLASOV EQUATION

We investigate the semiclassical limit from the semirelativistic 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) = -1 |x|a, a \in (max{ 2d-2,-1}, d -2], d \in {2, 3}, and -\in R, with the convention K(x) = -log(|x|) if a = 0. For mixed states, we show convergence in the Schatten norms with an explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [E. Dietler, S. Rademacher, and B. Schlein, J. Stat. Phys., 172 (2018), pp. 398--433], where the case of smooth pot entials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions.