The Hartree equation for infinitely many particles. I. Well-posedness theory

We show local and global well-posedness results for the Hartree equation $$i{\mathrm{\partial }}_t\gamma =[-\mathrm{\Delta }+w\ast {\rho }_{\gamma },\gamma ],$$ where $\gamma$ is a bounded self-adjoint operator on $L^2({\mathbb{R}}^d)$, ${\rho }_{\gamma }(x)=\gamma (x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $\gamma (0)$ is assumed to be a perturbation of a translation-invariant state ${\gamma }_f=f(-\mathrm{\Delta })$ 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 $\gamma (t)$, counted relatively to the stationary state ${\gamma }_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-\mathrm{\Delta })$. 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.