Continuous dependence for NLS in fractional order spaces

Abstract

For the nonlinear Schrödinger equation $i{u}_t+\mathrm{\Delta }u+\lambda |u{|}^{\alpha }u=0$ in ${\mathbb{R}}^N$, local existence of solutions in $H^s$ is well known in the $H^s$-subcritical and critical cases $0<\alpha ⩽4/(N-2s)$, where $0<s<\min \{N/2,1\}$. 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, 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, $\alpha ⩾1$ then the flow is locally Lipschitz.