Continuous dependence for NLS in fractional order spaces

Cazenave, Thierry; Fang, Daoyuan; Han, Zheng

doi:10.1016/j.anihpc.2010.11.005

Cazenave, Thierry ; Fang, Daoyuan ; Han, Zheng

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 135-147

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Résumé

For the nonlinear Schrödinger equation $i u_{t} + Δ u + λ {| u |}^{α} u = 0$ in $ℝ^{N}$ , local existence of solutions in $H^{s}$ is well known in the $H^{s}$ -subcritical and critical cases $0 < α ⩽ 4 / (N - 2 s)$ , 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, $α ⩾ 1$ then the flow is locally Lipschitz.

MR Zbl

DOI : 10.1016/j.anihpc.2010.11.005

Classification : 35Q55, 35B30, 46E35
Keywords: Schrödinger's equation, Initial value problem, Continuous dependence, Fractional order Sobolev spaces, Besov spaces

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     author = {Cazenave, Thierry and Fang, Daoyuan and Han, Zheng},
     title = {Continuous dependence for {NLS} in fractional order spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {135--147},
     year = {2011},
     publisher = {Elsevier},
     volume = {28},
     number = {1},
     doi = {10.1016/j.anihpc.2010.11.005},
     mrnumber = {2765515},
     zbl = {1209.35124},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.11.005/}
}

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Cazenave, Thierry; Fang, Daoyuan; Han, Zheng. Continuous dependence for NLS in fractional order spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 135-147. doi: 10.1016/j.anihpc.2010.11.005

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