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- W2048427680 abstract "In the 2-d setting, given an H 1 solution v(t) to the linear Schrödinger equation i∂ t v + Δ v = 0, we prove the existence (but not uniqueness) of an H 1 solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂ t u + Δu -|u| p-1 u = 0 for nonlinear powers 2 < p < 3 and the existence of an H 1 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)‖ H 1 → 0 as t → + ∞. This is a partial result toward the existence of well-defined continuous wave operators H 1 → H 1 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." @default.
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- W2048427680 date "2010-03-01" @default.
- W2048427680 modified "2023-09-26" @default.
- W2048427680 title "ASYMPTOTICALLY LINEAR SOLUTIONS IN H1 OF THE 2-D DEFOCUSING NONLINEAR SCHRöDINGER AND HARTREE EQUATIONS" @default.
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- W2048427680 doi "https://doi.org/10.1142/s0219891610002049" @default.
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