Global existence for 2D nonlinear Schrödinger equations via high–low frequency decomposition method

We study global existence of solutions for the Cauchy problem of the nonlinear Schrödinger equation iut + Δu = |u|2mu in the 2 dimension case, where m is a positive integer, m 2. Using the high–low frequency decomposition method, we prove that if 10m−6 10m−5 < s <1 then for any initial value ϕ ∈ Hs(R2), the Cauchy problem has a global solution in C(R,Hs(R2)), and it can be split into u(t) = eitΔϕ + y(t), with y ∈ C(R,H1(R2)) satisfying y(t) H1 (1 + |t |) 2(1−s) (10m−5)s−(10m−6)+ , where is an arbitrary sufficiently small positive number. © 2005 Elsevier Inc. All rights reserved.