The Cauchy problem for the fractional nonlinear Schro<spacing diaeresis>dinger equation in Sobolev spaces

We investigate the Cauchy problem for the fractional nonlinear Schro<spacing diaeresis>dinger (fNLS) equation. First, we establish the local well-posedness in the fractional Sobolev spaces H gamma with gamma >= 0 for the fNLS equation under less regularity assumptions for the nonlinear term than previous work. Based on the local well-posedness result, we then prove that the fNLS equation is globally wellposed in H gamma with gamma >= 0 if 4 sigma/d <= nu <= nu c(gamma) and the initial data is sufficiently small.