The defocusing energy-supercritical Hartree equation

Abstract

In this paper, we study the global well-posedness and scattering problem for the energysupercritical Hartree equation \(iu_t + \Delta u - (|x|^{ - \gamma } * |u|^2 )u = 0\) with γ > 4 in dimension d > γ. We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, u ∈ Lt∞ (I; 547-2 (ℝd)) with \(s_c : = \tfrac{\gamma } {2} - 1 > 1 \), 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 Schrodinger 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.