Phase mixing for the Hartree equation and Landau damping in the semiclassical limit

This talk focuses on the long-time behaviour of the Hartree equation near translation-invariant steady states, under short-range interaction potentials satisfying the Penrose stability condition. Phase-mixing estimates will be presented, showing decay of the density and scattering of solutions in quantum Sobolev spaces. These results remain uniform in the semiclassical limit, where the Hartree equation converges to the classical Vlasov equation. This provides a quantum analogue of Landau damping from classical plasma physics and connects quantum and classical kinetic models within a unified framework.