In this paper, we study the global well-posedness and scattering problem for the energy -supercritical Hartree equation iut ＋ △u - （｜χ｜^-r＊ ｜u｜^2）u = 0 with r〉 4 in dimension d 〉r. We prove that if the sol...In this paper, we study the global well-posedness and scattering problem for the energy -supercritical Hartree equation iut ＋ △u - （｜χ｜^-r＊ ｜u｜^2）u = 0 with r〉 4 in dimension d 〉r. We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, u ∈ Lt^∞（I;Hx^sc（R^d）） with Sc ：= x/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 SchrSdinger 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.展开更多
文摘In this paper, we study the global well-posedness and scattering problem for the energy -supercritical Hartree equation iut ＋ △u - （｜χ｜^-r＊ ｜u｜^2）u = 0 with r〉 4 in dimension d 〉r. We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, u ∈ Lt^∞（I;Hx^sc（R^d）） with Sc ：= x/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 SchrSdinger 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.
