Global derivation of the 1D Vlasov-Poisson equation from quantum many-body dynamics with screened Coulomb potential

Abstract

We study the 1D quantum many-body dynamics with a screened Coulomb potential in the mean-field setting. Combining the quantum mean-field, semiclassical, and Debye length limits, we prove the global derivation of the 1D Vlasov-Poisson equation. We tackle the difficulties brought by the pure state data, whose Wigner transforms converge to Wigner measures. We find new weighted uniform estimates around which we build the proof. As a result, we obtain, globally, stronger limits, and hence the global existence of solutions to the 1D Vlasov-Poisson equation subject to such Wigner measure data, which satisfy conservation laws of mass, momentum, and energy, despite being measure solutions. This happens to solve the 1D case of an open problem regarding the conservation law of the Vlasov-Poisson equation raised in [18] by Diperna and Lions.