Continuous dependence for NLS in fractional order spaces

By T. Cazenave, Daoyuan Fang, Zheng-chao Han. 2010

71 citations recorded by the source corpus.

Abstract For the nonlinear Schrodinger 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 → H s . If, in addition, α ⩾ 1 then the flow is locally Lipschitz.

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Citations in the corpus

On well posedness for the inhomogeneous nonlinear Schr\"odinger equation (2016, background): “In addition, Cazenave-Weissler [5] and recently Cazenave-Fang-Han [3] showed that the IVP for the NLS is locally well posed in H(R ) if 0 < α ≤ 4 N−2s and 0 < s < N 2 , moreover the local solution extends globally in time for small initial data.”
Composition Operators on Function Spaces with Fractional Order of Smoothness (Harmonic Analysis and Nonlinear Partial Differential Equations) (2011, background): “to [28], [34], [49], where boundedness and continuity of composition operators are treated in connection with the Schrödinger equation.”
Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schr\"odinger Equations (2015, background): “[27] to be continuous from H(R) to C([0, T ];H(R)).”