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 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 , where ϵ is an arbitrary sufficiently small positive number.