The defocusing energy-supercritical Hartree equation

In this paper, we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+.u.(|x|...|u|2)u=0 with . > 4 in dimension d > .. We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, u . L .t (I; 547-2 (.d)) with sc:=.2.1>1, then u is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation (NLW) and nonlinear Schrödinger equation (NLS). We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.