Long-time dynamics of Vlasov--Hartree systems across the Coulomb threshold

Abstract

We study the three-dimensional Vlasov--Hartree system with Coulomb and inverse-power interactions. For small, regular, localized data, we prove global well-posedness and determine long-time dynamics. The main result concerns the Coulomb case, where the coupling persists at leading order through reciprocal asymptotic corrections: the fermionic distribution scatters along logarithmically corrected free characteristics set by the asymptotic bosonic profile, while the bosonic wave gains a logarithmic phase set by the asymptotic fermionic density. We also treat longer- and shorter-range interactions. Our proof combines Hamiltonian methods for the Vlasov equation with purely physical-space vector-field methods for dispersive equations. For stronger long-range interactions, we introduce a symplectic transformation and a regularized effective-phase cancellation, which remove leading nonintegrable interactions while preserving Hamiltonian structure and avoiding derivative loss.