Smoothing for the fractional Schroedinger equation on the torus and the real line
Smoothing for the fractional Schroedinger equation on the torus and the real line
复制标题
环面和实线上分数阶薛定谔方程的平滑
DOI：
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发表时间：
2017
影响因子：
1.1
通讯作者：
N. Tzirakis
中科院分区：
文献类型：
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作者：
M. Erdogan;T. Gurel;N. Tzirakis
本文研究了环面上和真实的直线上的三次分数阶非线性薛定谔方程。结合规范形和限制范数方法，我们证明了解的非线性部分比初始数据更光滑。我们的方法适用于聚焦和散焦的非线性。在全色散（NLS）的情况下，在环面上，增益是全导数，而在真实的线上，我们得到具有$epsilon$损失的导数平滑。我们的结果降低了最近Kappeler等人关于周期散焦三次NLS的一个定理的正则性要求，并将其推广到聚焦情形和真实的直线。在散焦情形下，我们还得到了整体光滑解的高阶Sobolev范数的估计。
In this paper we study the cubic fractional nonlinear Schrodinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a full derivative, while on the real line we get a derivative smoothing with an $epsilon$ loss. Our result lowers the regularity requirement of a recent theorem of Kappeler et al. on the periodic defocusing cubic NLS, and extends it to the focusing case and to the real line. We also obtain estimates on the higher order Sobolev norms of the global smooth solutions in the defocusing case.