2008/05/19 by Justin Holmer, Holmer, Justin, Nikolaos Tzirakis +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.0805.2925
arxiv created 2009/07/03 · arxiv updated 2009/12/01
In the 2-d setting, given an H1 solution v(t) to the linear Schrödinger equation i∂t v +Δv =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 (2008) to a strategy devised by Nakanishi (2001). For the Hartree equation, we prove a new correlation estimate following the method of Colliander-Grillakis-Tzirakis (2008).