Modul: MAT675 PDE and Mathematical Physics

## Semiclassical limit from Hartree to Vlasov equation

Laurent Lafleche talk

Speaker invited by: Dr. Chiara Saffirio

**Date:** 16.05.19 **Time:** 14.00 - 15.00 **Room:** Y27H12

The Vlasov equation describes the evolution of a system of particles in interaction at a mesoscopic scale. Its counterpart in quantum mechanics is the Hartree equation and it can be proved that it converges in some sense to the Vlasov equation when the Planck constant \hbar becomes negligible. In this talk, I will present how this convergence can be quantitatively measured by introducing the Wigner transform and semiclassical versions of the Wasserstein-Monge-Kantorovitch distance and the kinetic Lebesgue norms. One of the key step to reach this result is the propagation in time of semiclassical moments and weighted Schatten norms of the solution, which implies the boundedness of the spatial density of particles. This can be proved by using the formal analogies between the Vlasov equation and the density operator formulation of quantum mechanics.