ASYMPTOTICALLY LINEAR SOLUTIONS IN H1 OF THE 2-D DEFOCUSING NONLINEAR SCHRöDINGER AND HARTREE EQUATIONS

Abstract

In the 2-d setting, given an H1 solution v(t) to the linear Schrödinger equation i∂tv + Δ v = 0, we prove the existence (but not uniqueness) of an H1 solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂t u + Δu -|u|p-1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H1 solution u(t) to the defocusing Hartree equation i∂t u + Δu -(|x|-γ⋆|u|2)u = 0 for interaction powers 1 < γ < 2, such that ‖u(t) - v(t)‖H1 → 0 as t → + ∞. This is a partial result toward the existence of well-defined continuous wave operators H1 → H1 for these equations. For NLS in 2-d, such wave operators are known to exist for p ≥ 3, while for p ≤ 2 it is known that they cannot exist. The Hartree equation in 2-d only makes sense for 0 < γ < 2, and it was previously known that wave operators cannot exist for 0 < γ ≤ 1, while no result was previously known in the range 1 < γ < 2. Our proof in the case of NLS applies a new estimate of Colliander–Grillakis–Tzirakis to a strategy devised by Nakanishi. For the Hartree equation, we prove a new correlation estimate following the method of Colliander–Grillakis–Tzirakis.