Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations

Abstract

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < α< 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-α}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3α}{4(α+ 1)} < s < \frac {2-α}4$ in the sense that the flow map is not locally uniformly continuous.