论文标题
1D立方散落的分散方程的全球解决方案：第一部分
Global solutions for 1D cubic defocusing dispersive equations: Part I
论文作者
论文摘要
本文致力于与立方非线性的一维NLS问题的一般类别。在近年来，获得此类问题的全球时间解决方案的散射问题引起了很多关注，并且在许多模型中，许多模型都证明了许多全球适应性的结果，假设初始数据既是\ emph {small} and \ emph {entabized}。但是，除了完全可以整合的情况外，对于小但非定位的初始数据，尚无此类结果。 在本文中，我们介绍了一种新的非扰动方法，以证明$ l^2 $初始数据的全局良好和散射，该数据是\ emph {small}，但\ emph {nonlopalized}。我们的主要结构假设是我们的非线性是\ emph {defocusing}。但是，我们不认为我们的问题有任何确切的保护法。我们的方法是基于对摩拉维兹估计的互动概念的强大重新解释，该估计是由I-Team开发的。 在散射方面，我们证明我们的全球解决方案满足了全球$ l^6 $ strichartz估计和双线性$ l^2 $界限。这是伽利亚的不变结果，即使对于古典散热焦点NLS来说，这也是新的。在那里，通过扩展我们的结果也承认了一个大数据。
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 is both \emph{small} and \emph{localized}. However, except for the completely integrable case, no such results have been known for small but non-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 is \emph{small} but \emph{non-localized}. Our main structural assumption is that our nonlinearity is \emph{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. There, by scaling our result also admits a large data counterpart.
