The nonlinear Hartree equation for N particles is given as a system of equations for N orthonormal functions, and it can be reformulated as a single operator-valued PDE in the Heisenberg picture. This reformulated equation has a richer structure than the system. For instance, it has stationary solutions, which are simply Fourier multiplier, having infinite many particle number. They include some physically important examples describing the thermal equilibria. In this talk, we discuss the local and global well-posedness problem on perturbations from these stationary solutions. For a suitable local theory, we employ Strichartz estimates for density functions, rephrased as Strichartz estimates for orthonormal functions. For global well-posedness, we make use of the relative entropy and the relative free energy. This problem has been first introduced by Lewin and Sabin recently in 2013, and then improved in several aspects in the joint work with Thomas Chen and Natasa Pavlovic at University of Texas at Austin.