On the blow up phenomenon for the mass critical focusing Hartree equation in R

Here f(u) = λ ( V ∗|u|2 ) u, V (x) = |x|−γ , 0 < γ < d, and ∗ denotes the convolution in Rd. If λ > 0, we call the equation (1.1) defocusing; if λ < 0, we call it focusing. This equation describes the mean-field limit of many-body quantum systems; see, e.g., [6], [7] and [36]. An essential feature of Hartree equation is that the convolution kernel V (x) still retains the fine structure of micro two-body interactions of the quantum system. By contrast, NLS arise in further limiting regimes where two-body interactions are modeled by a single real parameter in terms of the scattering length. In particular, NLS cannot provide effective models for quantum system with long-range interactions such as the physically important case of the Coulomb potential V (x) ∼ |x|−(d−2) in d ≥ 3, whose scattering length is infinite.