Global existence for 2D nonlinear Schrodinger equations via high-low frequency decomposition method

We study global existence of solutions for the Cauchy problem of the nonlinear Schrodinger equation iu(t) + Delta u = vertical bar u vertical bar(2m)u 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 psi is an element of H-S(R-2), the Cauchy problem has a global solution in C(R, H-S(R-2)), and it can be split into u(t) = e(it Delta)psi + y(t), with y is an element of C(R, H-1 (R-2)) satisfying vertical bar vertical bar(t)vertical bar vertical bar(H1) <= (1 + vertical bar t vertical bar)(2(1-s))/((10m-5)s-(10m-6))+epsilon, where epsilon is an arbitrary sufficiently small positive number. (c) 2005 Elsevier Inc. All rights reserved.