ASYMPTOTICALLY LINEAR SOLUTIONS IN H¹OF THE 2-D DEFOCUSING NONLINEAR SCHRöDINGER AND HARTREE EQUATIONS

Abstract

In the 2-d setting, given an H 1 solution v(t) to the linear Schrödinger equation i∂ t v + ∆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 + ∆u − |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 equationThis 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 [4] to a strategy devised by Nakanishi [15]. For the Hartree equation, we prove a new correlation estimate following the method of [4].

Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy

“…Analogous estimates hold for the case of the Hartree equation iu t + ∆u = λ(|x| −γ ⋆ |u| 2 )u when 0 < γ < n, n ≥ 2. For the details, see [37]. We should point out that for 0 < γ ≤ 1 scattering fails for the Hartree equation, [35], and thus the estimates given in [37] for n ≥ 2 cover all the interesting cases.…”

“…For mass sub-critical solutions, scattering even in the energy space is a very hard problem, and is probably false. Nevertheless, two particle Morawetz estimates have been used for the problem of the existence (but not uniqueness) of the wave operator for mass subcritical problems, [37]. We have already mentioned their implementation to the hard problem of energy critical solutions in [2], [31], and [17].…”

The nonlinear Schrödinger equations with combined nonlinearities of power-type and Hartree-type

“…By using Hölder, Hardy-Littlewood-Sobelov inequality, we have 10) and from (3.3),(3.5) and (3.10), we have…”

Randomized final-data problem for systems of nonlinear Schrödinger equations and the Gross–Pitaevskii equation

“…On the final-data problem, the first author proved [31] existence of a solution u scattering in L 2 (R d ) for any final-data u + ∈ L 2 (R d ) for p ∈ (2/d, 4/d) and d ≥ 3, which was extended to d = 2 by Holmer and Tzirakis [22] in the case u + ∈ H 1 (R 2 ) and λ ≥ 0. However, the uniqueness of the solution u for a given final-state u + is an open question in those results, since the proof relies crucially on a compactness argument.…”