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We show local and global well-posedness results for the Hartree equation$$i\partial_t\gamma=[-\Delta+w*\rho_\gamma,\gamma],$$ where $\gamma$ is abounded self-adjoint operator on $L^2(\R^d)$, $\rho_\gamma(x)=\gamma(x,x)$ 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$\gamma_f=f(-\Delta)$ which describes a quantum system with an infinite numberof particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac andBose-Einstein gases at positive temperature. Global well-posedness follows fromthe conservation of the relative (free) energy of the state $\gamma(t)$,counted relatively to the stationary state $\gamma_f$. We indeed use a generalnotion of relative entropy, which allows to treat a wide class of stationarystates $f(-\Delta)$. Our results are based on a Lieb-Thirring inequality atpositive density and on a recent Strichartz inequality for orthonormalfunctions, which are both due to Frank, Lieb, Seiringer and the first author ofthis article.

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