Optimal Local Well-Posedness for the Periodic Derivative Nonlinear Schrodinger Equation
Optimal Local Well-Posedness for the Periodic Derivative Nonlinear Schrodinger Equation
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DOI：
10.1007/s00220-020-03898-8
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发表时间：
2020-11-14
影响因子：
2.4
通讯作者：
Yue, Haitian
中科院分区：
文献类型：
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作者：
Deng, Yu;Nahmod, Andrea R.;Yue, Haitian
我们证明了周期导数的非线性薛定谔方程是L-2临界的，在尺度类似H-S(T)的傅立叶-勒贝格空间中证明了其局部适定性。我们的结果是最优的，因为它涵盖了整个亚临界状态。特别地，我们通过改进GrunRock和Herr(SIAMJ Math anal39(6)：1890-1920,2008)的结果填补了次临界理论中的现有空白，该结果建立了像S和Gt；1/4的H-S(T)尺度的傅里叶-勒贝格空间的局部适定性。我们通过对解的结构的精细分析和适当函数空间的一个适应的非线性子流形的构造来获得这一结果。综合起来，我们可以为给定的次临界数据构建唯一的解决方案。这一建设性程序的灵感来自于Gubinelli等人发展的拟控制分布理论。(论坛数学派，3：75,2015)和Catellier和Chouk(Ann Probab 46(5)：2621-2679,2018)。然而，我们的证明和结果纯粹是决定论的。
We prove local well-posedness for the periodic derivative nonlinear Schrodinger equation, which is L-2 critical, in Fourier-Lebesgue spaces which scale like H-s (T) for s > 0. Our result is optimal in the sense that it covers the full subcritical regime. In particular we close the existing gap in the subcritical theory by improving the result of Grunrock and Herr (SIAM J Math Anal 39(6):1890-1920, 2008), which established local well-posedness in Fourier-Lebesgue spaces which scale like H-s (T) for s > 1/4. We achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space. Together these allow us to construct the unique solution to the given subcritical data. This constructive procedure is inspired by the theory of para-controlled distributions developed by Gubinelli et al. (Forum Math Pi, 3:75, 2015) and Catellier and Chouk (Ann Probab 46(5):2621-2679, 2018) in the context of stochastic PDE. Our proof and results however, are purely deterministic.