Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation
Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation
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DOI：
10.1016/j.aim.2019.02.001
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发表时间：
2018-02
影响因子：
1.7
通讯作者：
B. Dodson;J. Luhrmann;Dana Mendelson
中科院分区：
文献类型：
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作者：
B. Dodson;J. Luhrmann;Dana Mendelson
本文研究了四维空间中散焦三次非线性薛定谔方程的柯西问题，建立了随机初值在Hx S(R4)中的几乎处处适定和条件几乎处处散射的证明，证明的主要内容是引入了一个研究伴随强迫三次非线性薛定谔方程的泛函框架，它的灵感来自于研究薛定谔映射问题的某些函数空间，它基于Strichartz空间以及局部光滑化、非齐次局部光滑化和极大值函数空间的变种.此外，我们证明了随机径向对称初值在H×S(R4)中的一个几乎必然的散射结果。
We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in H x s (R 4) with 1 3< s< 1. The main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear Schrödinger equation, which is inspired by certain function spaces used in the study of the Schrödinger maps problem, and is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing, and maximal function spaces. Additionally, we prove an almost sure scattering result for randomized radially symmetric initial data in H x s (R 4) with 1 2< s< 1.