Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov–Poisson

Abstract In this paper we establish almost-optimal stability estimates in quantum optimal transport pseudometrics for the semiclassical limit of the Hartree dynamics to the Vlasov–Poisson equation, in the regime where the solutions have bounded densities. We combine Golse and Paul’s method from [Arch Ration Mech Anal 223:57–94, 2017], which uses a semiclassical version of the optimal transport distance and which was adapted to the case of the Coulomb and gravitational interactions by the second author in [J Stat Phys 177:20–60, 2019], with a new approach developed by the first author in [Arch Ration Mech Anal 244:27–50, 2022] to quantitatively improve stability estimates in kinetic theory.