Continuous dependence for NLS in fractional order spaces

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation $i{u}_t+\Delta u+\lambda |u{|}^{\alpha }u=0$ in ${\mathbb{R}}^N$, in the $H^s$-subcritical and critical cases $0<\alpha \le 4/(N-2s)$, where $0<s<N/2$. 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 $s=0,1,2$.

Corrected typos. Simplified section 4. Results unchanged

References in corpus (1)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case

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