Scattering of the energy-critical NLS with dipolar interaction

Abstract

In this paper, we investigate the global well-posedness and H 1 scattering theory for a 3d energy-critical Schrödinger equation under the influence of magnetic dipole interaction λ 1 |u| 2 u + λ 2 (K * |u| 2 )u, where K is the dipole-dipole interaction kernel. Our proof of global well-posedness result is based on the argument of Zhang [23]. Moreover, adopting the induction of energy technique of Killip-Oh-Pocovnicu-Visan [20], we obtain a condition for scattering. 1 2 u(λ 2 t, λ −1 x) preserves both the equation and the energy. Global well-posedness and scattering theory for the energy critical and mass critical NLS has been intensively studied in the last years; see Colliander-Keel-Staffilani-Takaoka-Tao [10], Tao-Visan-Zhang [21], J. Bourgain [7], B. Dodson [12], Kenig-Merle [14], Killip-Visan [15] and references therein for more details.