The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

We show local and global well-posedness results for the Hartree equation(Formula presented.) where γ is a bounded self-adjoint operator on (Formula presented.), ρ_γ(x) = γ(x, x) and w is a smooth short-range interaction potential. The initial datum γ(0) is assumed to be a perturbation of a translation-invariant state γ_f = f(−Δ) which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi–Dirac and Bose–Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state γ(t), counted relatively to the stationary state γ_f. We indeed use a general notion of relative entropy, which allows us to treat a wide class of stationary states f(−Δ). Our results are based on a Lieb–Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.