Continuous dependence for NLS in fractional order spaces

We consider the Cauchy problem for the nonlinear Schr"odinger equation $i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0$ in ${\mathbb{R}}^N$, in the $H^s$-subcritical and critical cases $0<\alpha \le 4\mathrm{/}(N-2s)$, where $0<s<N\mathrm{/}2$. Local existence of solutions in $H^s$ is well known. However, even though the solution is constructed by a fixed-point technique, continuous dependence in $H^s$ does not follow from the contraction mapping argument. In this paper, assuming furthermore $s<1$, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous $H^s\to H^s$. If, in addition, $\alpha \ge 1$ then the flow is Lipschitz. This completes previously known results concerning the cases $s=0,1,2$.