Global solutions for 1D cubic defocusing dispersive equations: Part I
Global solutions for 1D cubic defocusing dispersive equations: Part I
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DOI：
10.1017/fmp.2023.30
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
M. Ifrim;D. Tataru
中科院分区：
文献类型：
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作者：
M. Ifrim;D. Tataru
摘要本文研究了一类具有立方非线性的一维非线性最小二乘问题。近年来，在初始数据既小又局部化的假设下，获得这类问题的散布全局解的问题引起了人们的广泛关注，许多模型的全局适定性结果已被证明。然而，除了完全可积的情况外，对于小但不一定是局部化的初始数据，还没有已知这样的结果。本文介绍了一种新的非摄动方法来证明$L^2$初值的全局适定性和散射性。我们的主要结构假设是，我们的非线性是散焦的。然而，我们并不假定我们的问题有任何确切的守恒定律。我们的方法是基于对莫拉维茨估计的交互作用想法的强有力的重新解释，该想法是由I-Team在近20年前开发的。在散度方面，我们证明了我们的整体解既满足整体$L^6$Strichartz估计，又满足双线性$L^2$界。这是一个伽利略不变的结果，即使对于经典的散焦三次NLS也是新的。1通过标度，我们的结果也允许大量的数据对应。
Abstract This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both small and localized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data. In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for $L^2$ initial data which are small and nonlocalized. Our main structural assumption is that our nonlinearity is defocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.1 There, by scaling, our result also admits a large data counterpart.