Global-in-time Semiclassical Regularity for the Hartree-Fock Equation

For arbitrarily large times $T>0$, we prove the uniform-in-$\mathrm{\hslash }$ propagation of semiclassical regularity for the solutions to the Hartree$\text{unicode}x2013$Fock equation with singular interactions of the form $V(x)=\pm \mathrm{\mid }x{\mathrm{\mid }}^{-a}$ where $a\in (0,\frac{1}{2})$. As a byproduct of this result, we extend to arbitrarily long times the derivation of the Hartree$\text{unicode}x2013$Fock and the Vlasov equations from the many-body dynamics provided in [J. Chong, L. Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].