Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

Understand

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\geq 3$, including the case of a Coulomb singularity in dimension $d=3$.

• This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small.
• As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects.