Energy-critical Hartree equation with harmonic potential for radial data

✍ Scribed by Haigen Wu; Junyong Zhang

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 907 KB
• Volume: 72
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2009.11.026

✦ Synopsis

In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions (n ≥ 5) and show the global well-posedness and scattering theory in the space Σ = H 1 ∩F H 1 . We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao's approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory.