Well-posedness and ill-posedness for the cubic fractional Schrödinger equations
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- Well-posedness and ill-posedness for the cubic fractional Schrödinger equations
- Cho, Yonggeun; Hwang, Gyeongha; Kwon, Soonsik; Lee, Sanghyuk
- Issue Date
- AMER INST MATHEMATICAL SCIENCES
- DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v.35, no.7, pp.2863 - 2880
- We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices 1 < α < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in Hs for s ≥ 2-α/4. This is shown via a trilinear estimate in Bourgain's Xs,b space. We also show that non-periodic equations are ill-posed in Hs for 2-3α/4(α+1) < s < 2-α/ 4 in the sense that the flow map is not locally uniformly continuous.
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