The Cauchy problem for the fractional nonlinear Schrödinger equation in Sobolev space

We investigate the Cauchy problem for the fractional nonlinear Schrödinger (fNLS) equation. First, we establish the local well-posedness in the fractional Sobolev spaces $H^{\gamma}$ with $\gamma\geq 0$ 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 well-posed in $H^{\gamma}$ with $\gamma\geq 0$ if $4\sigma/d \leq \nu \leq \nu_c (\gamma )$ and the initial data is sufficiently small.